# Fast Outflows and Luminous He II Emission in Dwarf Galaxies with AGN

Weizhe Liu (刘伟哲) Steward Observatory, University of Arizona, 933 N. Cherry
Ave., Tucson, AZ 85721, USA Weizhe Liu<EMAIL_ADDRESS>Sylvain Veilleux
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
Joint Space-Science Institute, University of Maryland, College Park, MD 20742,
USA  Gabriela Canalizo Department of Physics and Astronomy University of
California Riverside, 900 University Avenue, CA 92521, USA  Todd M. Tripp
Department of Astronomy, University of Massachussetts, Amherst, MA 01003, USA
David S. N. Rupke Department of Physics, Rhodes College, Memphis, TN 38112,
USA  Archana Aravindan Department of Physics and Astronomy, University of
California, Riverside, 900 University Ave, Riverside CA 92521, USA  Thomas
Bohn Department of Physics and Astronomy University of California Riverside,
900 University Avenue, CA 92521, USA  Fred Hamann  Department of Physics and
Astronomy, University of California, Riverside, CA 92507, USA  Christina M.
Manzano-King Department of Physics and Astronomy University of California
Riverside, 900 University Avenue, CA 92521, USA

###### Abstract

While stellar processes are believed to be the main source of feedback in
dwarf galaxies, the accumulating discoveries of AGN in dwarf galaxies over
recent years arouse the interest to also consider AGN feedback in them. Fast,
AGN-driven outflows, a major mechanism of AGN feedback, have indeed been
discovered in dwarf galaxies and may be powerful enough to provide feedback to
their dwarf hosts. In this paper, we search for outflows traced by the
blueshifted ultraviolet absorption features in three dwarf galaxies with AGN
from the sample examined in our previous ground-based study. We confirm
outflows traced by blueshifted absorption features in two objects and
tentatively detect an outflow in the third object. In one object where the
outflow is clearly detected in multiple species, photoionization modeling
suggests that this outflow is located $\sim$0.5 kpc from the AGN, implying a
galactic-scale impact. This outflow is much faster and possesses higher
kinetic energy outflow rate than starburst-driven outflows in sources with
similar star formation rates, and is likely energetic enough to provide
negative feedback to its host galaxy as predicted by simulations. Much broader
($\sim$4000 km s-1) absorption features are also discovered in this object
which may have the same origin as that of broad absorption lines in quasars.
Additionally, strong He ii $\lambda$1640 emission is detected in both objects
where the transition falls in the wavelength coverage, and is consistent with
an AGN origin. In one of these two objects, blueshifted He ii $\lambda$1640
emission line is clearly detected, likely tracing a highly-ionized AGN wind.

††facilities: HST(COS), HST(WFC3)††software: Astropy (Astropy Collaboration et
al., 2013, 2018), BayesVP (Liang & Chen, 2014), CALCOS
(https://github.com/spacetelescope/calcos), ChiantiPy (Dere et al., 1997,
2019), CLOUDY (Ferland et al., 1998), LMFIT (Newville et al., 2016), NumPy
(Harris et al., 2020), SciPy (Virtanen et al., 2020).

##  1 Introduction

Stellar processes have long been considered the main source of negative
feedback in dwarf galaxies (e.g. Larson, 1974; Veilleux et al., 2005; Heckman
& Thompson, 2017; Martín-Navarro & Mezcua, 2018). However, it is still debated
whether such stellar feedback is effective enough to reproduce all the related
properties of the dwarf galaxies we see today (e.g. Garrison-Kimmel et al.,
2013; McQuinn et al., 2019). Recent studies have revealed hundreds of active
galactic nuclei (AGN) in dwarf galaxies through multi-wavelength observations
(see the recent reviews by Greene et al., 2019; Reines, 2022). AGN activity
can release a tremendous amount of energy, and AGN feedback has been widely
accepted as a critical mechanism to regulate the formation and evolution of
massive galaxies (e.g., Di Matteo et al., 2005; Booth & Schaye, 2009; Fabian,
2012). Therefore, it is interesting to also consider the possible impact of
AGN feedback in dwarf galaxies. Moreover, low-z dwarf galaxies likely possess
properties close to those of moderate AGN host galaxies in the early universe
(e.g., Matthee et al., 2023; Maiolino et al., 2023). Such AGN feedback may
also regulate the growth of (super)massive black hole seeds and first galaxies
(e.g. Silk, 2017).

Evidence of AGN feedback in dwarf galaxies has emerged in recent years.
Specifically, hints of star formation quenching induced by AGN feedback (Penny
et al., 2018) and lower-than-expectation global HI content (Bradford et al.,
2018) have both been reported in dwarf galaxies with AGN. Similar evidence on
such feedback is also seen in several latest observational studies (Davis et
al., 2022; Yang et al., 2023; Kawamuro et al., 2023). From the theoretical
perspective, analytic analysis (e.g. Silk, 2017; Dashyan et al., 2018) and new
simulations (e.g. Koudmani et al., 2019, 2021; Lanfranchi et al., 2021) have
all pointed out the possibly significant effects of AGN feedback in dwarfs. In
addition to the negative feedback discussed above that suppresses star
formation, AGN-driven outflows may also trigger star formation activities
within dwarf galaxies (Schutte & Reines, 2022).

Recently, a sample of 29 dwarf galaxies with AGN (stellar mass
log(M⋆/$\mathrm{M}_{\odot}$) $<$ 10.0) was observed with Keck optical
spectroscopy (Manzano-King et al., 2019; Manzano-King & Canalizo, 2020) and
NIR spectroscopy (Bohn et al., 2021), revealing fast outflows in 9 of them.
This sample was chosen from dwarf galaxies at z $<$ 0.05 in SDSS showing
properties in the optical consistent with AGN activity (i.e., broad Balmer
lines, Seyfert-like line ratios on the BPT/VO87 (Baldwin et al., 1981;
Veilleux & Osterbrock, 1987) diagrams and/or highly-ionized He ii line
emission; Reines et al., 2013; Moran et al., 2014). Follow-up integral field
spectroscopy (IFS) of 8 of the 29 dwarf galaxies confirmed the existence of
rapid outflows in 7 of them (Liu et al., 2020, L20 hereafter). These outflows
are primarily driven by the AGN, and are powerful enough to provide feedback
for their host galaxies in a way similar to outflows of more luminous AGN with
massive hosts. Additionally, such outflows are more powerful than those
observed in a matched sample of non-AGN star-forming dwarf galaxies with
similar galaxy properties (Aravindan et al., 2023).

While the results from the optical emission lines are tantalizing, they only
probe the relatively denser part of the outflows as the emission line strength
is proportional to the electron density squared. Absorption lines, instead,
depend on column densities linearly and are thus more sensitive probes of the
full extent of outflows captured along the line of sight. Moreover,
blueshifted absorption features are unambiguous signatures of the outward
motion of gas located in front of the source of continuum radiation. Rest-
frame far-ultraviolet (FUV) spectroscopy provides access to multiple strong
absorption features spanning a broad range of ionization states, which is
essential for finer measurements and/or tighter upper limits on the physical
conditions of outflows (e.g., more robust ionization correction).

Such FUV spectroscopy has been used extensively to probe ionized and neutral
gas outflows triggered by AGN (e.g. Crenshaw & Kraemer, 2012) and/or star
formation (Heckman et al., 2015; Heckman & Borthakur, 2016), in galaxies with
starbursts (e.g. Martin et al., 2015) or recently-quenched star formation
(e.g. Tripp et al., 2011). To evaluate the AGN feedback via fast outflows in
dwarf galaxies with the same criterion, a sample of dwarf galaxies with
primarily AGN-driven outflows needs to be examined with the same scrutiny.

In this paper, we present the results from a pilot HST/COS FUV spectroscopic
study of three dwarf galaxies with evidence of AGN-driven outflows from L20.
In Section 2, the data sets, the physical properties of the sources measured
from the HST/COS and ancillary data, and the data reduction procedures are
described. The detection and characterization of absorption line-traced
outflows and thus AGN feedback within the sample are presented in Section 3.
The analysis of the strong He ii $\lambda$1640 emission lines detected in
these objects is described in Section 4. Throughout the paper, we assume a
$\Lambda$CDM cosmology with $H_{0}$ = 69.3 km s-1 Mpc-1, $\Omega_{\rm m}$ =
0.287, and $\Omega_{\rm\Lambda}=0.713$ (Hinshaw et al., 2013).

##  2 HST and Ancillary Data

###  2.1 HST/COS Observations and Data Reduction

We observed the three FUV-brightest dwarf galaxies with AGN-driven outflows
from the sample studied in L20 (see Table 1 for object names and their short
names). The targets were observed with HST/COS through cycle 27 program PID
15915 (PI: W. Liu). The corresponding observations can be accessed via
http://dx.doi.org/10.17909/9rjd-2819 (catalog 10.17909/9rjd-2819). The spectra
of our targets were obtained in TIME-TAG mode through the Primary Science
Aperture (PSA) using the medium resolution FUV grating, G160M. Four focal
plane offset positions were adopted to reduce the impact of fixed-pattern
noise associated with the detector for all targets, and the wavelength setting
was adjusted according to the redshift of the target and was selected to cover
major absorption line/emission line features including C ii $\lambda$1335, C
iv $\lambda$$\lambda$1548, 1550, Si ii $\lambda$1526, and Si iv
$\lambda$$\lambda$1394, 1403. The spectral resolutions of the observations
vary among the three objects: objects J0906$+$56 and J0954$+$47 are point
sources in the NUV acquisition images and their spectral resolution thus
remains the nominal value of $\sim$15-20 km s-1. The third object J1009$+$26,
is extended in the NUV acquisition image and the spectral resolution is thus
degraded. Following the method outlined in Henry et al. (2015), we used the
NUV acquisition image and the point source line spread function to derive the
spectral resolution. Specifically, we first created a one-dimensional profile
in the dispersion direction of the acquisition image, by summing the pixels in
the cross dispersion direction. We then convolved this profile with the point
source LSF to approximate the LSF for this spatially extended object and
estimated its spectral resolution to be $\sim$30-40 km s-1.

The raw data were processed and combined by the CALCOS pipeline v3.3.10.
CALCOS corrects the data for instrumental effects, assigns a vacuum wavelength
scale, and extracts flux-calibrated spectra. It applies a heliocentric
correction to the final x1d files for each exposure, and combines the
individual exposures to a single spectra.

Figure 1: Archival HST F606W images for sources J0906$+$56 (a) and J0954$+$47
(b), and SDSS composite image for J1009$+$26 (c). In each panel, the red
horizontal bar indicates the corresponding spatial scale in arcseconds.

###  2.2 Ancillary Data

In addition to the HST/COS data, we retrieve key physical properties of our
sources from L20. Moreover, the circular velocity of the objects, $v_{circ}$
are derived based on their stellar mass, adopting the baryonic Tully-Fisher
relation calibrated in Reyes et al. (2011) as
log$(v_{cir})=0.278$log$(M_{*})-0.67$, with root-mean-square (RMS) residuals
of 0.1 dex in $v_{circ}$. This relation was derived using spatially-resolved
2D gas kinematics in low-redshift emission-line galaxies where
$v_{cir}=\sqrt{2(v^{2}_{rot}+2\sigma^{2})}$. Here $v_{rot}$ and $\sigma$ are
the rotation speed and mean value of the line-of-sight velocity dispersion,
respectively. These results are summarized in Table 1.

We also retrieve the archival HST F606W images (PID 13943; PI A. Reines) and
SDSS images (Abazajian et al., 2009) for our sources (Fig. 1). The relevant
HST data can be accessed via http://dx.doi.org/10.17909/yp2h-yb46 (catalog
10.17909/yp2h-yb46). The morphologies of our three sources are different.
While J0906$+$56 is likely a late-type galaxy with possible spiral arms,
J0954$+$47 and J1009$+$26 show no sign of spiral arms and are more consistent
with early-type galaxies.

Table 1: Properties of the Objects

Name | Short Name | Redshift | log(Mstellar) | log($v_{circ}$) | R50 | log(L[OIII]) | log(LAGN,[OIII]) | log(LAGN,FUV) | SFR
---|---|---|---|---|---|---|---|---|---
|  |  | [$\mathrm{M}_{\odot}$] | [km s-1] | [kpc] | [erg s-1] | [erg s-1] | [erg s-1] | [$M_{\sun}\ yr^{-1}$]
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
SDSS J090613.75$+$561015.5 | J0906$+$56 | 0.0467 | 9.4 | 1.9$\pm{0.1}$ | 1.5 | 41.15${}^{+0.01}_{-0.01}$ | 43.7 | 42.9 |  $<$0.3
SDSS J095418.16$+$471725.1 | J0954$+$47 | 0.0327 | 9.1 | 1.9$\pm{0.1}$ | 2.0 | 41.36${}^{+0.02}_{-0.02}$ | 43.9 | 42.8 |  $<$0.3
SDSS J100935.66$+$265648.9 | J1009$+$26 | 0.0145 | 8.8 | 1.8$\pm{0.1}$ | 0.7 | 40.48${}^{+0.01}_{-0.01}$ | 43.0 | 42.0 |  $<$0.1

Note. — Column (1): SDSS name of the object; Column (2): Short name of the
object used in this paper; Column (3): Redshift of the target measured from
the stellar fit to the spectrum integrated over the KCWI data cube; Column
(4): Stellar mass in unit of solar mass and in logarithm from the NASA Sloan
Altas (http://www.nsatlas.org/); Column (5): Circular velocity in logarithm
derived from stellar mass adopting calibration in Reyes et al. (2011); Column
(6): Half-light radius from the NSA, in unit of kpc; Column (7): Total [O iii]
$\lambda$5007 luminosity based on the observed total [O iii] $\lambda$5007
fluxes within the field of view of the KCWI data without extinction
correction, in units of erg s-1; Column (8): Bolometric AGN luminosity, based
on the extinction-corrected [O III] luminosity, in units of erg s-1. Column
(9): Bolometric AGN luminosity, based on the monochromatic continuum
luminosity $L_{1320}$ from HST/COS data and the CLOUDY AGN SED. See Appendix A
for more details; Column (10): Upper limit on the star formation rate (SFR)
based on the extinction-corrected [O ii] $\lambda$$\lambda$3726,3729 flux from
the KCWI data, in units of $M_{\sun}\ yr^{-1}$. Here we assume that 1$/$3 of
the [O ii] $\lambda$$\lambda$3726,3729 emission is from the star formation
activity, following Ho (2005).

Figure 2: The fit to the C iv absorption + emission line complex in
J0906$+$56\. Top: The best-fit C iv emission line doublet model (red) with two
Gaussian components to recover the unabsorbed emission line profile. The
fitting windows are marked on the best-fit model in cyan; Bottom: The best-fit
model (red) of the absorption features normalized to the best-fit emission
line model shown in the top panel. The data is in black and the error is in
green. The individual Voigt profile components are shown as blue dashed lines.
The velocity is with respect to the systemic velocity of the bluer line at the
redshift listed in Table 1.

Figure 3: The C ii $\lambda$1335, Si iv $\lambda$1394, Si ii $\lambda$1526,
and C iv $\lambda$1550 absorption features in J0954$+$47, from top to bottom.
Left: The best-fit continuum/emission line model (red) in the vicinity of the
absorption feature. The fitting windows are marked on the best-fit model in
cyan. For C ii, the continuum is fit with a linear function. For Si iv and Si
ii, the broad absorption trough in the vicinity of each absorption feature is
fit with a 2nd-order polynomial. For C iv, the intrinsic C iv
$\lambda$$\lambda$1548, 1550 emission line doublet is fit with two Gaussian
profiles with the same centroid velocity and velocity dispersion, and a fixed
flux ratio of 2. The centroid velocity and velocity dispersion are fixed to
those of the He ii $\lambda$1640 emission in this object. The UV continuum is
fit with a linear function to featureless regions beyond the plotting range of
the figure. Note that the blueshifted, broad C iv absorption trough and the
possible, redshifted, broad C iv emission line wing are not considered in our
fit due to the large uncertainties. Right: The best-fit models (red) of the
absorption features normalized by the best-fit continuum/emission line models
shown in the left panel. The spectra are in the rest-frame of the
corresponding absorption features. The individual Voigt profile components are
shown as blue dashed lines. In the fit, the velocity and $b$ parameter (line
width) of individual components for all three lines (C ii $\lambda$1335, Si iv
$\lambda$1394, Si ii $\lambda$1526) are tied together. The C iv $\lambda$1550
is excluded from the fit due to the large uncertainties in modeling intrinsic
emission line$+$continuum profile, but the normalized absorption line profile
is shown for visual comparison with the other three absorption features. The
velocity is with respect to the systemic velocity at the redshift listed in
Table 1.

##  3 Outflows Traced by Blueshifted Absorption Features

###  3.1 Detection and Characterization of Outflows

To characterize the absorption features in individual objects, we first fit
the continua and unabsorbed emission line profiles in them with low-order
polynomials and Gaussian functions convolved with the HST/COS line-spread
function (LSF)111Tabulated on the HST/COS website https://hst-
docs.stsci.edu/cosihb, respectively. Next, the observed spectra are normalized
with the best-fit continuum and intrinsic emission line models to obtain the
normalized absorption line profiles. We then fit these normalized profiles
with multiple velocity components with Voigt profiles, where the normalized
intensity can be written as:

$\displaystyle I(v)=\prod\\{1-C_{f}[1-e^{-\tau(v)}]\\}$ (1)
$\displaystyle\tau(v|N,b,z)=N\sigma_{0}f\Phi(v|b,z)$ (2)

$\tau$, $v$, $N$, $b$, $z$, $f$, and $\sigma_{0}$ are the optical depth,
velocity, ion column density, Doppler parameter, redshift, oscillator
strength, and cross section, respectively. $\Phi(v|b,z)$ is the normalized
Voigt profile. The adopted atomic parameters are taken from Morton (2003). In
all fits, the models are further convolved with the LSF, and the spectra are
binned by 3 pixels for Nyquist sampling. We adopted a customized software
built on the non-linear least-squares fit implemented in LMFIT (Newville et
al., 2016) to search for the best-fit model for the absorption features. In
our software, a velocity component (with Voigt profile) is added to the model
if the Bayesian Information Criterion (BIC; Ivezić et al., 2014) decreases,
and this process stops when the minimum BIC value is found. The parameter
uncertainties are calculated based on the output from the python package
LMFIT, which are the standard error from the estimated covariance matrix.

None of the absorption features shows statistically significant evidence of
partial covering (e.g. non-black saturated profiles). Specifically, while the
bottoms of C iv absorption troughs in J0906$+$56 (Fig. 2) are above zero, they
are consistent with the noise level of the observations. The best-fit model
also gives a partial covering factor of 1 (i.e., no partial covering based on
the current S/N of our spectrum). As a result, we fix the covering fraction to
1 in our fits to all objects and equation (1) is thus simplified to
$I(v)=\prod[e^{-\tau(v)}]$.

Overall, objects J0906$+$56 and J0954$+$47 show clear evidence of outflows
traced by absorption features. In object J1009$+$26, the detection is
tentative as the blueshift of the absorption features is modest with large
uncertainties. In the following, we describe the characteristic of the
absorption features and fitting details for each object.

In J0906$+$56, only blueshifted C iv $\lambda$$\lambda$1548, 1550 absorption
features are detected on top of the broad, C iv emission lines (Fig. 2 and
Appendix B). We fit the featureless continuum with a linear function and the
intrinsic C iv emission line doublet with two Gaussian profiles, using fitting
windows with no absorption signatures. The Gaussian profiles share the same
centroid velocity and velocity dispersion, and their flux ratio is fixed to 2,
the theoretical expectation in the optically thin limit. We also explore the
case where the flux ratio of the emission line doublet is allowed to vary, but
the final results remain similar. For the absorption feature, a total of two
velocity components are required by the best-fit. The final best-fits for the
continuum/emission line profile and the absorption line profiles are shown in
Fig. 2.

In J0954$+$47, blueshifted absorption features with similar profiles are seen
in C ii $\lambda$1335, C iv $\lambda$$\lambda$1548, 1550, Si ii $\lambda$1526,
Si iv $\lambda$$\lambda$1394, 1403 and possibly O i $\lambda$1302 (Fig. 3 and
Appendix B). Additionally, much broader ($\sim$4000 km s-1) absorption troughs
are seen for both C iv and Si iv doublets (but not for other lower ionization
transitions; e.g., see Fig. 4 for C iv as an example). The line widths of
these broad troughs fall in the typical range of those of broad absorption
lines (BAL) seen in luminous quasars (e.g. Weymann et al., 1991). If these
broad troughs are indeed BAL features as in luminous quasars, it would be the
first BAL ever discovered in a dwarf AGN to our knowledge, with a balnicity
(Weymann et al., 1991) of $\sim$740 km s-1. Nevertheless, these absorption
troughs are possibly accompanied by redshifted broad emission line features
reaching $\sim$2000 km s-1 (with peak S/N $\sim$ 3). The entire features may
thus also be P-Cygni-like profiles and instead trace a spatially unresolved
stellar wind in this object.

To further characterize the narrow absorption features (i.e., excluding the
potential BALs), we first focus on the C ii $\lambda$1335, Si iv
$\lambda$1394, and Si ii $\lambda$1526 lines. These absorption features are
either not or only mildly contaminated by nearby emission and/or broad
absorption features (Appendix B) and thus the derived normalized absorption
line profiles are relatively more robust. Specifically, the continuum near C
ii $\lambda$1335 is fit with a linear function. The broad absorption trough in
the vicinity of Si iv $\lambda$1394 and Si ii $\lambda$1526 are both fit with
2nd-order polynomials. Next, for the C iv $\lambda$1550 absorption feature, we
aim to obtain an approximate profile of it as it is blended with the nearby C
iv doublet emission, C iv $\lambda$1548 absorption, and broad C iv BAL-like
trough. The intrinsic C iv $\lambda$$\lambda$1548, 1550 emission line is fit
with two Gaussian profiles with the same centroid velocity and velocity
dispersion, and a fixed flux ratio of 2. The fitting window is chosen as the
presumably pure C iv emission-line wing, and the centroid velocity and
velocity dispersion are fixed to those of the He ii $\lambda$1640 emission
line. As we only care about the spectral range of $\gtrsim-$500 km s-1 of C iv
$\lambda$1550 where the contribution from the broad C iv BAL-like trough
should be trivial, this broad trough is thus not modeled in the fit. Finally,
for the C iv $\lambda$1548 and Si iv $\lambda$1394 absorption features, the
nearby emission/absorption features are more complex than those near C iv
$\lambda$1550 and our attempt to constrain the intrinsic absorption line
profiles results in large uncertainties. These two features are thus not
considered in the following analysis.

We then obtain normalized absorption line profiles for C ii, Si iv and Si ii
based on the best-fit continuum/emission line profiles above, and fit them
simultaneously where the centroid velocities and velocity dispersions (i.e.,
Doppler parameters) of the same velocity components are tied together. A total
of four velocity components are required by the best-fit. The final best-fits
for the continuum/emission line profiles and the absorption line profiles are
shown in Fig. 3.

In J1009$+$26, the overall featureless continuum is fit with a linear
function. The unabsorbed C iv emission line profile is modeled in the same way
as that for J0954$+$47, except that the line width of C iv is not fixed to
that of He ii but set as a free parameter instead. The recovered C iv
absorption features are relatively weak and poorly constrained due to the
large uncertainties of the unabsorbed C iv emission line profile.

We then examine the normalized absorption features closely and find that the
blueshift of absorption features is in general modest (Fig. 5 and Appendix B).
Si iv $\lambda$$\lambda$1394, 1403 and Si ii $\lambda$1526 absorption features
are mildly blueshifted with $v_{50}$ $\sim-$20 km s-1 when each feature is fit
individually with a single Voigt profile (Here $v_{50}$ is the velocity at the
location that accumulates 50% of the total equivalent width (EW) of the entire
absorption feature). Blueshifted C iv $\lambda$$\lambda$1548, 1550 absorption
features at similar velocities may also be present, despite the large
uncertainties resulting from the modeling of unabsorbed C iv emission line.
However, the C ii $\lambda$1335 absorption feature is not blueshifted when fit
alone with a Voigt profile and is detected only at the systemic velocity with
a relatively large uncertainty ($v_{50}$ $\sim 0\pm{30}$ km s-1). Moreover,
when Si ii, Si iv and C ii absorption features are fit together with the same
centroid velocity and Doppler parameter, the blueshift is small with best-fit
$v_{50}$ $\sim-15\pm{15}$ km s-1. Additionally, the absorption line profiles
are smooth and broad (Doppler parameter b$\sim$190 km s-1), which are
different from the profiles typically observed in intervening and ISM
absorbers in star-forming galaxies (e.g., Werk et al., 2013; Savage & Sembach,
1996) but similar to intrinsic AGN absorbers often associated with outflows
(e.g., Hamann & Ferland, 1999). Overall, we cannot completely rule out the
possibility that the absorption features are tracing the kinematically
quiescent gas at/near systemic velocity and thus the outflow in this object is
only marginally detected.

In our final fit to the normalized absorption features, we adopt the
simultaneously fit to C ii, Si iv and Si ii where the centroid velocities and
Doppler parameters of the same velocity components are tied together. We find
that only one velocity component is required by the best-fit, and the final
best-fits for the continuum/emission line profiles and the absorption line
profiles are shown in Fig. 5.

The final results from the best-fits of absorption features for all three
objects are summarized in Table 2.

Figure 4: Zoom-in view on the potential C iv BAL feature in J0954$+$47\. The
zero velocity of C iv $\lambda$1548 corresponds to the redshift listed in
Table 1 and is denoted by the vertical black line. The best-fit continuum with
a linear function is indicated by the blue dashed line. The red horizontal bar
denotes the breadth of the potential BAL feature.

Figure 5: The C ii $\lambda$1335, Si iv $\lambda$$\lambda$1394, 1403, Si ii
$\lambda$1526, and C iv $\lambda$$\lambda$1548, 1550 absorption features in
J1009$+$26, from top to bottom. Left: The best-fit continuum/emission line
model (red) in the vicinity of the absorption feature. The fitting windows are
marked on the best-fit model in cyan. The featureless continuum is fit with a
linear function to selected wavelength windows marked in cyan. The intrinsic C
iv $\lambda$$\lambda$1548, 1550 emission line doublet is fit with two Gaussian
profiles with the same centroid velocity and velocity dispersion, and a fixed
flux ratio of 2. The location of the corresponding transitions at the systemic
velocity are indicated by blue vertical dashed lines. Right: The best-fit
models (red) of the absorption features normalized by the best-fit
continuum/emission line models shown in the left panel. The C iv feature is
excluded from the fit due to the large uncertainties in modeling the
unabsorbed emission line profile, but the normalized absorption line profile
is shown for visual comparison with the other three absorption features. In
the fit, the velocity and $b$ parameter (line width) of individual components
for all lines are tied together. For Si iv and C iv, only the bluer line of
the doublet is shown for a zoom-in view of the absorption line profile. The
velocity is with respect to the systemic velocity at the redshift listed in
Table 1.

###  3.2 The Galactic-scale Outflow in J0954$+$47

The blueshifted absorption features of various ions in J0954$+$47 share
similar velocity distributions (see Fig. 3), tracing the same fast outflow
within this object (The potential C iv and Si iv BALs shown in Fig. 4 and
described in Section 3.1 are not considered here due to the lack of
constraints on its radial distance based on our data). The lower limit on the
ratio of C ii $\lambda$1335 and C ii* $\lambda$1336 lines (C ii* $\lambda$1336
undetected) also provides tight constraint on the electron density and thus
the radial distance of the outflow. In the remainder of this section, we take
advantage of these constraints to derive the physical properties of the
outflow in J0954$+$47 and evaluate the power source and impact of the outflow.
Note that we refer to the entire absorption feature as the “entire outflow”,
and the most blueshifted velocity component in the best-fit model shown in
Fig. 3 as the “high-v outflow” hereafter. Such analysis is not reliable and
thus not carried out for J0906$+$56 and J1009$+$26, since the outflow is only
detected in C iv for the former and is only tentatively detected for the
latter.

Table 2: Properties of the Outflows

Object | Absorber | Comp. | $v_{50}$ | $b$ |  $log(N_{ion}$) |  $log(N_{H}$) | $R$ | $log(\dot{m}_{out})$ | $log(\dot{p}_{out})$ | $log(\dot{E}_{out})$
---|---|---|---|---|---|---|---|---|---|---
|  |  | [km s-1] | [km s-1] | [$cm^{-2}$] | [$cm^{-2}$] | [kpc] | [$M_{\sun}\ yr^{-1}$] | [$dynes$] | [$erg\ s^{-1}$]
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
J0906$+$56 | C iv | entire | $-130\pm{40}$ |  $180\pm{20}$, $60\pm{10}$ | $15.3\pm{0.1}$ | … | … | … | … | …
high-v | $-320\pm{30}$ | $180\pm{20}$ | $14.4\pm{0.1}$ | … | … | … | … | …
J0954$+$47 | C ii | entire | $-260\pm{20}$ | $100\pm{10}$, $60\pm{12}$, $85\pm{30}$, $50\pm{10}$ | $15.0\pm{0.1}$ | $19.7\pm{0.1}$ | $0.49\pm{0.04}$ | $-0.1\pm{0.2}$ | $33.2\pm{0.2}$ | $40.6\pm{0.2}$
Si iv | $14.4\pm{0.1}$
Si ii | $14.7\pm{0.2}$
C ii | high-v | $-460\pm{10}$ | $100\pm{10}$ | $14.60\pm{0.04}$ | $19.20\pm{0.06}$ | $0.52\pm{0.03}$ | $-0.3\pm{0.3}$ | $33.2\pm{0.3}$ | $40.9\pm{0.3}$
Si iv | $13.90\pm{0.07}$
Si ii | $14.10\pm{0.15}$
J1009$+$26 | C ii | entire | $-15\pm{15}$ | $190\pm{10}$ | $14.9\pm{0.1}$ | … | … | … | … | …
Si iv | $14.2\pm{0.1}$
Si ii | $14.8\pm{0.1}$

Note. — Column (1): Object name; Column (2): Ion considered; Column (3):
Velocity component considered in the calculation. The “entire” label indicates
the values for the entire absorption feature. Specifically, we adopt this
$v_{50}$ as the outflow velocity for the calculation of mass, momentum and
kinetic energy outflow rate of the entire outflow in column (9)–(11). So these
are effectively average outflow rates. The “high-v” label indicates the
component with the highest outflow velocity in the best-fit model. Column (4):
Velocity at the location that accumulates 50% of the total equivalent width
(EW) of the entire absorption feature; Column (5): Doppler parameter from the
best-fits. For the “entire component”, we list the value for each individual
component separately; Column (6): Ion column density in logarithm; Column (7):
Hydrogen column density in logarithm derived from Section 3.2.1; Column (8):
Outflow radial distance in kpc derived from Section 3.2.1; Column (9): Mass
outflow rate in logarithm; Column (10): Momentum outflow rate in logarithm;
Column (11): Kinetic energy outflow rate in logarithm.

####  3.2.1 Photoionization Modeling

Figure 6: Phase plot showing the ionization solutions for the entire outflow
(left) and the high-v outflow (right) of J0954$+$47\. Each colored contour
represents the locus of models (outflow radial distance $R$, hydrogen column
density $N_{H}$) which predicts a ion column density consistent with the
observed value of that ion. The width of each locus corresponds to the
1-$\sigma$ uncertainty in the observed ion column density. The center of the
black cross indicates the best solution based on the $\chi^{2}\ $ minimization
and the size of the cross indicated the corresponding uncertainties. See
Section 3.2.1 for more details.

We adopt CLOUDY (C22.00 Ferland et al., 1998) to model the properties of both
the entire outflow and the high-v outflow adopting the measured column density
of the outflow components of C ii, Si ii, and Si iv $\lambda$1394, assuming
that they trace the same outflowing material. The O i $\lambda$1302 absorption
feature is too weak (S/N$\sim$2.5) to be measured robustly, and we do not
include it in our modeling described below.

We set up the modeling as follows: we assume that the ionizing spectrum is a
typical AGN continuum described in Section 6.2 of the CLOUDY
manual222https://gitlab.nublado.org/cloudy/cloudy/-/wikis/home, which gives
$f_{\nu}=\nu^{\alpha_{uv}}exp(-h\nu/kT_{BB})exp(-kT_{IR}/hv)+a\nu^{\alpha_{x}}$.
$T_{BB}$ is the temperature of the “Big Blue Bump” component of the AGN, a
rising power law with a high-energy exponential cut-off. $\alpha_{uv}$ is the
low-energy slope of the “Big Bump” component. $\alpha_{x}$ is the slope of the
X-ray component. The coefficient $a$ is adjusted to produce the correct
$\alpha_{OX}$. We normalize this AGN continuum at 1320 Å to the observed flux
density of the COS spectrum. We also set the X-ray to UV ratio, $\alpha_{OX}$,
to $-$1.9 as measured from Baldassare et al. (2017). We set the other three
free parameters, $T_{BB}$, $\alpha_{uv}$, and $\alpha_{x}$ provided by CLOUDY
to their default values (1.5$\times$105 K, $-$0.5 and $-$1, respectively,
which are typical for AGN). Two more parameters are needed for the modeling,
the electron density $n_{e}$ and metal abundance. Due to the absence of C ii*
$\lambda$1336 absorption, we obtain an upper limit of $\sim$ 10 cm-3 for
$n_{e}$ adopting the upper limit on the C ii*/C ii ratio333The required atomic
data are retrieved from CHIANTI atomic database for astronomical spectroscopy
via the Python interface ChiantiPy. On the other hand, we do not expect
$n_{e}$ to be much less than 10 cm-3, given that the typical $n_{e}$ for
outflows in star-forming dwarf galaxies is $\gtrsim$ 10 cm-3 (e.g. Xu et al.,
2022). Therefore, we set $n_{e}$ to 10 cm-3 in the following analysis. As for
the metal abundance, we set it to solar, the same as the values obtained from
the photoionization modeling of the optical emission lines (L20).

With the model set-up above, the column density of each absorption feature
(with its 1-$\sigma$ uncertainty) defines a locus in the 2D diagram of total
hydrogen column density $N_{H}$ versus outflow radial distance $R$444The
default independent variable in CLOUDY is the ionization parameter
$U=\frac{Q}{4\pi cn_{e}R^{2}}$. With a given $n_{e}$, we convert $U$ into R..
Following the same approach adopted in previous work (e.g. Borguet et al.,
2012; Arav et al., 2013), the best-fit total hydrogen column density $N_{H}$
and the outflow radial distance $R$ is then obtained by minimizing the
$\chi^{2}\ $:

$\displaystyle\chi^{2}=\sum_{i}(\frac{logN_{i,mod}-logN_{o,obs}}{logN_{i,obs}-log(N_{i,obs}\pm{\sigma_{i}})})^{2}$
(3)

where, for ion $i$, $N_{i,obs}$ and $N_{i,mod}$ are the observed and modeled
column densities, and $\sigma_{i}$ is the error in the measured column
density.

The best-fit $N_{H}$ and $R$ are $19.7\pm{0.1}$ cm-2 and $0.49\pm{0.04}$ kpc
for the entire outflow and $19.20\pm{0.06}$ cm-2 and $0.52\pm{0.03}$ kpc for
the high-v outflow, respectively. The 1-$\sigma$ errors are estimated as the
full x and y extent of the overlapping region of the three loci in Fig. 6.
These results are listed in Table 2.

####  3.2.2 Energetics

With the $N_{H}$ and $R$ obtained above, the mass, momentum and kinetic energy
outflow rate can then be calculated, using the equations:

$\displaystyle\dot{m}_{out}=\Omega N_{H}\mu m_{p}Rv$ (4)
$\displaystyle\dot{p}_{out}=\Omega N_{H}\mu m_{p}Rv^{2}$ (5)
$\displaystyle\dot{E}_{out}=\frac{1}{2}\Omega N_{H}\mu m_{p}Rv^{3}$ (6)

In the equations above, the outflow velocity $v$ is set to $v_{50}$ (velocity
at half the equivalent width) of the best-fit absorption line profile. 555Note
that we are effectively calculating an average for the entire outflow by
adopting this definition of outflow velocity, and assuming that terms
$v(r)n(r)r$, $[v(r)]^{2}n(r)r$ and $[v(r)]^{3}n(r)r$ are constant in the
integral $\int_{0}^{4\pi}\int_{0}^{R}v(r)n(r)rdrd\Omega$,
$\int_{0}^{4\pi}\int_{0}^{R}[v(r)]^{2}n(r)rdrd\Omega$, and
$\int_{0}^{4\pi}\int_{0}^{R}[v(r)]^{3}n(r)rdrd\Omega$ to obtain
$\dot{m}_{out}$, $\dot{p}_{out}$, and $\dot{E}_{out}$, respectively. This
leads to the $N_{H}Rv$, $N_{H}Rv^{2}$, and $N_{H}Rv^{3}$ terms in equations
(4)–(6).. $\Omega$ is the solid angle subtended by the outflow as seen from
the center. While $\Omega$ cannot be measured directly from our data, the
outflow likely has a wide opening angle: the outflow seen in the optical
emission lines, which should be physically connected with the outflow seen in
absorption lines, favors such geometry (L20). In addition, no evidence of
partial covering is present in the absorption line profiles. Therefore, we set
$\Omega=4\pi$ in our calculations. The final results are recorded in Table 2.
Note again that as stated in Section 3.2.1, it is possible that the actual
$n_{e}$ of the outflow is smaller than the 10 cm-3 adopted, so the outflow
radial distance and outflow rates could be even larger than listed in Table 2.

The spatial extent and energetics of the outflow derived from the absorption
features are consistent with those measured from the blueshifted [O iii]
$\lambda$5007 emission line (L20; $R<1.6$ kpc and log $\dot{E}_{out}>39.6$ erg
s-1). This suggests that the blueshifted absorption and emission features are
tracing the same galactic outflow in this object, as assumed in the above
calculations.

Figure 7: Outflow velocity ($v_{50}$) as a function of SFR (left panels) and
circular velocity (right panels) based on the overall absorption feature
(entire outflow; upper panels) and the highest velocity component in the best-
fit (high-v outflow; lower panels) of J0954$+$47\. In the upper panels, the
small diamonds with error bars are the Si ii outflows in star-forming galaxies
from Chisholm et al. (2015). In these two panels, the values of $v_{50}$ are
all measured from the overall absorption features. In the bottom panels, the
outflows in the CLASSY sample from Xu et al. (2022) are shown in small
diamonds with error bars. For each object, $v_{50}$ is the median value of all
absorption features detected. In these two panels, $v_{50}$ is measured from
the highest velocity component of the best-fit model. The black dashed lines
in the two right panels indicate the expected escape velocity $v_{esc}\simeq
3v_{circ}$. In the two panels on the left, the data points are colored by
stellar mass. In the two panels on the right, the data points are colored by
SFR (star-forming sources) or the upper limits on SFR (J0954$+$47).

####  3.2.3 Comparison with Starburst-driven Outflows

Star-formation-driven outflow is a popular feedback mechanism in dwarf
galaxies. To understand the nature and impact of the fast outflow in
J0954$+$47, we compare its properties with those of star-formation-driven
outflows in low-z galaxies observed by HST/COS.

As shown in Fig. 7, we plot the outflow velocities as functions of SFR666The
upper limit on SFR for J0954$+$47 is derived from the [O ii]
$\lambda$$\lambda$3726,3729 luminosity (based on Ho, 2005, see L20 for
details), while the SFR of star-forming galaxies from Xu et al. (2022) are
obtained from Berg et al. (2022a), which are based on UV$+$optical spectral
energy distribution (SED) fitting, and the SFR of star-forming galaxies from
Chisholm et al. (2015) are obtained adopting both IR and UV data. There may be
systematic differences between these three types of SFR measurements, but we
do not expect these discrepancies to be large enough to change our results
qualitatively. and circular velocities for both J0954$+$47 and those from
Chisholm et al. (2015) or Xu et al. (2022). We only include sources with UV
radii $r_{50}<1.5$″ from Xu et al. (2022, i.e., red filled circles in their
Fig. 9), as the UV light of these sources lies within the HST/COS aperture,
which allows for reliable measurements of their global outflow properties. In
Chisholm et al. (2015), the outflow velocity is measured from the overall
best-fit Si ii $\lambda$1260 profile. In Xu et al. (2022), the outflow
velocity is measured as the median $v_{50}$ of the blueshifted components in
the best-fit model for all transitions detected, including O i $\lambda$1302,
C ii $\lambda$1335, Si ii 1190, 1193, 1260, 1304, 1526, Si iii $\lambda$1206,
and Si iv $\lambda$$\lambda$1394, 1403. For a fair comparison, therefore, we
show two outflow velocities for J0954$+$47: the upper panels show $v_{50}$ of
the overall absorption line profile as in Chisholm et al. (2015), and the
bottom panels show $v_{50}$ of the high-v outflow, similar to that defined in
Xu et al. (2022). In all panels, the data points are colored according to
their SFR or stellar mass.

With a SFR of $<$ 0.3 $M_{\sun}\ yr^{-1}$, the outflow in J0954$+$47 is much
faster than those in star-forming galaxies with similar SFR from both Chisholm
et al. (2015) and Xu et al. (2022). The presence of the AGN in this dwarf
galaxy appears to boost the outflow velocity. This is in agreement with the
results in Aravindan et al. (2023) where they find a larger outflow velocity
in dwarf galaxies with AGN than star-forming dwarf galaxies based on an
integral field spectroscopy study of emission lines. This is also consistent
with the results from the simulations in Koudmani et al. (2019) where the
presence of AGN feedback in dwarf galaxies leads to higher outflow velocity
than pure stellar feedback.

Given the circular velocity ($v_{circ}$) of $\sim$ 70 km s-1 (based on the
stellar mass of J0954$+$47), the outflow in J0954$+$47 is also faster than
those star-formation-driven ones from Chisholm et al. (2015) with similar
circular velocities and is among the fastest outflow seen among the star-
formation-driven ones from Xu et al. (2022). This suggests that, for a given
gravitational potential, AGN can at least drive an outflow as fast as those
seen in dwarf galaxies with extreme starbursts.

Moreover, in Fig. 8, we further compare the mass loading factor (defined as
the ratio of mass outflow rate to SFR), momentum outflow rate and kinetic
energy outflow rate of the outflow in J0954$+$47 with those of star formation-
driven outflows in low-z galaxies, taking sources from Xu et al. (2022) as an
example.

For a given SFR, the mass loading factor and momentum outflow rate of
J0954$+$47 are comparable to those of star-forming galaxies, whereas the
kinetic energy outflow rate is larger than those of star-forming galaxies by
$\sim$1 dex. These are again largely consistent with the results from the IFS
study presented in Aravindan et al. (2023) on optical emission lines, where
they find higher mass, momentum and kinetic energy outflow rates in dwarf
galaxies with AGN than star-forming ones, and the difference increases from
the mass rate to momentum rate and is the largest in the energy rate. Our
results also broadly agree with the results from the simulations in Koudmani
et al. (2019): while they find no significant difference in mass loading
factors between the simulation run with both AGN and supernovae feedback and
the one with only supernovae feedback, the thermal energy outflow rate in the
former is significantly larger than that in the latter by up to $\sim$5 dex.
These results are in concordance with the picture that, at a given SFR,
compared to stellar feedback, the AGN feedback deposits more energy to heat
the ISM and thus prevents the cooling of gas while having a similar impact on
expelling the gas out. However, this is based on only one dwarf galaxy with
AGN and a larger sample is needed before any robust conclusion can be drawn.

For a given stellar mass, while the mass loading factor of J0954$+$47 is still
comparable to the maximum values in star-forming galaxies, the momentum and
kinetic energy outflow rates of J0954$+$47 are close to the lowest values
observed in the star-forming galaxies. This is likely caused by the extreme
starburst activities in the sources from Xu et al. (2022) that are not typical
for z$\sim$0 dwarf galaxies like J0954$+$47\. The SFR of galaxies from Xu et
al. (2022) with similar stellar mass to J0954$+$47 are $\sim$1–2 dex larger
than both the upper limit of SFR in J0954$+$47 and the average SFR at z$\sim$0
(See the left panel of Fig. 3 in Berg et al., 2022b). Instead, the range of
SFR in the Xu et al. (2022) sample is comparable to those star-forming
galaxies at z$\sim$1–2.5, which may represent the most extreme cases of
stellar feedback, rarely seen in z$\sim$0 star-forming dwarf galaxies. The
much higher SFR in their sample thus allows for much higher energy output to
drive outflows more powerful than the presumably AGN-driven outflow in
J0954$+$47\. Therefore, at a given stellar mass, the most extreme stellar
feedback may still play a more important role than AGN feedback in dwarf
galaxies, if J0954$+$47 is a representative case for AGN feedback in dwarf
galaxies. However, this is again based on only one dwarf galaxy with AGN, and
it would be interesting to further look for more extreme AGN activities in
dwarf galaxies at higher redshift that may provide stronger feedback.

Figure 8: Mass loading fatcor $\eta$ (left column), momentum outflow rate
(middle column) and kinetic energy outflow rate (right column) as a function
of SFR (top row) and stellar mass (bottom row) for J0954$+$47 (red and blue
filled circles) and star-forming galaxies from Xu et al. (2022) (brown
crosses). The red and blue filled circles represents the average value for the
entire outflow and the value for the high-v outflow in J0954$+$47,
respectively. Figure 9: Ratio of kinetic energy outflow rate to AGN luminosity
as a function of AGN luminosity for the average value of the entire outflow
(red) and for the high-v outflow (blue) of J0954$+$47\. Also shown in black
are the X-ray and FUV absorber-traced outflows in nearby Seyferts from
Crenshaw & Kraemer (2012). Note that in the bottom left corner is the outflow
in the archetype dwarf Seyfert 1, NGC 4395. The horizontal dashed line at 0.5%
is the prediction for effective AGN feedback from Hopkins & Elvis (2010).

Figure 10: Momentum (left) and kinetic energy (right) outflow rates versus
expected momentum (left) and energy (right) injection rates from the star
formation activity (see Section 3.2.4) for the average value of the entire
outflow (red) and high-v outflow (blue) in J0954$+$47, and those in low-z
star-forming galaxies from Xu et al. (2022) (brown crosses). The dashed lines
in both panels indicate the 1:1 equality line.

####  3.2.4 Energy Source of the Outflow

In this section, we explore the possible energy source for the outflow in
J0954$+$47.

First, the AGN is powerful enough to easily drive the outflow, since the ratio
of the kinetic energy outflow rate to AGN luminosity, $\dot{E}_{out}/L_{AGN}$,
is $\sim$0.6% (the average for the entire outflow) or $\sim$1.2% (for the
high-v outflow), comparable to low-z AGN with outflow measured with the same
technique (Fig. 9; the LAGN value adopted here is derived based on the HST/COS
data and the AGN template from CLOUDY. See Appendix A for more details).

Second, as shown in Fig. 7 and stated at the end of Section 3.2.3, the outflow
is faster than those of outflows in dwarf galaxies with similar SFR, implying
that the presence of AGN is likely required to reach such a high outflow
velocity, which is also consistent with simulations (e.g., Koudmani et al.,
2019).

Finally, it is challenging but still possible in principle for star formation
activity alone to inject enough momentum and energy to the outflow. The total
momentum injection rate supplied by the star formation activity,
$\dot{p}_{in}=4.6\times 10^{33}SFR~{}[\mathrm{M}_{\odot}\ yr^{-1}]$ dynes,
which is the sum of the hot wind fluid driven by thermalized ejecta of massive
stars (Chevalier & Clegg, 1985) and radiation pressure (Murray et al., 2005).
Similarly, the total energy injection rate supplied by the star formation
activity is $\dot{E}_{in}=4.3\times 10^{41}SFR~{}[\mathrm{M}_{\odot}\
yr^{-1}]$ erg s-1. A SFR $<0.3$ $M_{\sun}\ yr^{-1}$ translates into a momentum
injection rate of $<1.4\times 10^{33}$ dynes and an energy injection rate of
$<1.3\times 10^{41}$ erg s-1. Given the measured momentum (kinetic energy)
outflow rate $\dot{p}_{out}$ $\simeq 1.6^{+1.6}_{-0.8}\times 10^{33}$ dynes
($\dot{E}_{out}$ $\simeq 7.9^{+7.9}_{-4.0}\times 10^{40}$ erg s-1), the star
formation activity is capable of driving the outflow if it can transfer the
momentum and energy to the outflow at high efficiencies ($\gtrsim$60% – 230%
for momentum and $\gtrsim$30% – 120% for kinetic energy). While possible,
these are not typical for nearby star-forming galaxies: as shown in Fig. 10,
for the majority of the star-forming galaxies in Xu et al. (2022), the ratios
of momentum (kinetic energy) outflow rates to the momentum (energy) injection
rates are lower than those of J0954$+$47\. Moreover, the momentum and energy
injection rates from star formation activity adopted above are uncertain. For
example, if adopting the estimates from Rupke et al. (2023) and scaling the
SFR that for our object, we obtain a $\dot{p}_{in}<4\times 10^{32}$ dynes and
$\dot{E}_{in}<8\times 10^{40}$ erg s-1 for SFR $<$ 0.3 $M_{\sun}\ yr^{-1}$.
The required efficiencies increase to $\gtrsim$210% – 810% for momentum and
$\gtrsim$50% – 200% for energy, making it even more difficult for the star
formation activity to drive the outflow.

Overall, AGN is likely the dominating energy source for the outflow.
Nevertheless, we cannot formally rule out the possibility that star formation
activity may also help launch the outflow in J0954$+$47.

####  3.2.5 The Impact of the Outflow

Given the measured large velocity of the outflow in J0954$+$47, some of the
outflowing gas may escape the host galaxy altogether. We estimate the escape
velocity as $v_{esc}^{2}=2v_{circ}^{2}[1+ln(r_{s}/R)]$ where the mass
distribution is modeled as a truncated isothermal sphere: $r_{s}$ is the
radius of the sphere, and R is the radius where the $v_{esc}$ is calculated as
$v_{esc}\simeq 3v_{circ}$ with $r_{s}/R\simeq 33$ as adopted in literature
(e.g., see Veilleux et al., 2020, for a brief discussion). The average
velocity of the entire outflow ($\sim$260 km s-1) and velocity of the high-v
outflow ($\sim$460 km s-1) in J0954$+$47 are comparable to, or significantly
larger than, $v_{esc}$ of the host galaxy ($\sim$220 km s-1). A large portion
($\sim$50% in terms of absorption line equivalent width) of the ionized gas
traced by the absorption may escape the system and help enrich the
circumgalactic medium.

The outflow also carries significant amounts of mass, momentum, and kinetic
energy. We obtain a mass loading factor (i.e., mass outflow rate over SFR) of
$\gtrsim$3.4 ($\gtrsim$1.8) for the average of the entire outflow (the high-v
outflow), suggesting that more gas is carried away by the outflow than that
fueling star formation. As shown in Fig. 9, the kinetic energy outflow rate to
AGN luminosity, $\dot{E}_{out}/L_{AGN}$, is $\sim$0.6% ($\sim$1.2%) for the
average of the entire outflow (the high-v outflow), which is comparable to
(larger than) the expectation from one representative simulation of AGN
feedback (e.g., Hopkins & Elvis, 2010) that only require $\sim$0.5% of the AGN
bolometric luminosity to be injected into the outflow for effective AGN
feedback.777According to Hopkins & Elvis (2010), AGN feedback will be
effective if $\eta\dot{m}\frac{M_{BH}}{<M_{BH}>}\sim 0.05f_{hot}$, where
$\eta=\dot{E}_{out}/L_{AGN}$, $\dot{m}$ is the Eddington ratio, ${M_{BH}}$ and
${<M_{BH}>}$ are the black hole mass and the expectation from the M-$\sigma$
relation, and $f_{hot}$ is the hot gas fraction which is typically $\sim$0.1
in ISM. If J0954$+$47 follows M-$\sigma$ relation
($\frac{M_{BH}}{<M_{BH}>}=1$), then $\dot{m}\sim$1\. Therefore, it leads to
$\dot{E}_{out}/L_{AGN}\sim 0.5\%$ for an effective feedback.
In addition, the prediction from such simulations are the kinetic coupling
efficiency, the fraction of the AGN bolometric luminosity coupled to gas near
the black hole. This is always larger than the observed
$\dot{E}_{out}/L_{AGN}$: Even for an energy-driven large-scale outflow, only a
portion of the original nuclear wind power ends up in kinetic form, where the
rest of the energy may be used up in doing work against the gravitational
potential and ambient pressure along the way (Harrison et al., 2018).
Therefore, the outflow in J0954$+$47 is likely powerful enough to provide
significant negative feedback to its dwarf host galaxy. Interestingly, the HST
image of this object (Fig. 1b) shows no evidence of spiral arms and suggests
no significant on-going star formation. The color of this source based on SDSS
images is also green ($u-r\sim 2$), suggesting that it is transitioning from a
blue star-forming galaxy to a red, passive galaxy (Manzano-King et al., 2019).
These properties are apparently consistent with the star formation quenching
caused by the negative feedback via the outflow.

Note that the kinetic energy outflow rate derived based on absorption lines
here is $\sim$10$\times$ larger than that obtained from previous [O iii]
$\lambda$5007 emission line studies ($log(\dot{E}_{out})\sim 39.6$ if the
outflow is spatially resolved; Liu et al., 2020). This discrepancy may be
largely due to the fact that [O iii] emission line misses the lower density
and/or lower-ionization outflowing gas which is more easily probed by the
absorption features examined in this paper. This emphasizes the importance of
probing outflows with different emission/absorption tracers to acquire a
complete census on the outflow energetics.

Figure 11: He ii $\lambda$1640 emission line profile compared with O iv]
$\lambda$1404, C iv $\lambda$1550, He ii $\lambda$4686, and [O iii]
$\lambda$5007 emission line profiles for J1009$+$26, respectively. All spectra
are continuum subtracted. The He ii $\lambda$1640 emission line is in its
original flux scale while all other emission lines are re-scaled to have the
same peak flux density as that of He ii $\lambda$1640\. The systemic velocity
is determined from the stellar kinematics based on the IFS data from L20 as
listed in Table 1.

##  4 He ii $\lambda$1640 Emission

###  4.1 Detection and Basic Properties

A strong He ii $\lambda$1640 emission line is detected in two objects,
J0954$+$47 (Appendix B)) and J1009$+$26 (Fig. 11 and Appendix B), consistent
with the AGN nature of these two objects, given the high ionization potential
of He+ ($\sim$54.4 eV). The equivalent widths of He ii are 2.71$\pm{0.14}$ Å
and 7.00$\pm{0.29}$ Å, respectively, again broadly consistent with the high
values seen in AGN (with a median He ii equivalent width of $\sim$8 Å; e.g.
Hainline et al., 2011; Cassata et al., 2013). The He ii/C iv flux ratios of
the two objects are $\sim$0.8 and $\sim$1.7, which fall in the range
($\sim$0.002$-$3) predicted by AGN photoionization models (e.g. Feltre et al.,
2016). In addition, the He ii/C iv flux ratios in the two objects are also
broadly consistent with values typically measured in Type 2 quasar/AGN ($\sim$
0.7; e.g. McCarthy, 1993; Corbin & Boroson, 1996; Humphrey et al., 2008;
Matsuoka et al., 2009).

The He ii lines in these two sources are narrow, with FWHM of 120$\pm{10}$ km
s-1 and 110$\pm{10}$ km s-1 for J0954$+$47 and J1009$+$26, respectively.
Therefore, these are not the broad He ii features observed in Wolf-Rayet
stars/galaxies (e.g. Abbott & Conti, 1987). Moreover, to our knowledge, for
star-forming galaxies in the nearby universe, narrow nebular He ii emission
lines have only been observed in metal-poor sources (e.g. Berg et al., 2016;
Senchyna et al., 2017), while the metallicity of our two objects, J0954$+$47
and J1009$+$26, are both close to solar values based on the modeling of
optical line ratios from L20. Additionally, the equivalent widths of those He
ii lines in nearby star-forming galaxies (up to $\sim$1.7 Å as reported in
those two references) are lower than those of J0954$+$47 and J1009$+$26.

Overall, the He ii $\lambda$1640 emission lines detected in our sample are
consistent with an AGN origin. A more detailed analysis that takes into
account all FUV, optical, and near-IR emission lines with various ionization
potentials will be the subject of a future paper. The hard ionization spectra
from dwarf galaxies with AGN like our objects may lead to high escape
fractions of ionizing photons, which, if present in abundance in the early
universe, may contribute to cosmic reionization (e.g. Madau & Haardt, 2015;
Robertson et al., 2015). Interestingly, the recent discovery of a population
of faint AGN at z $\sim$ 4–10 (e.g. Matthee et al., 2023; Harikane et al.,
2023; Maiolino et al., 2023; Übler et al., 2023) suggests that such dwarf AGN
may indeed be more common at the epoch of reionization than previously
expected.

###  4.2 Blueshifted He ii $\lambda$1640 Emission Tracing A Highly-ionized
Outflow in J1009$+$26?

An interesting characteristic of the He ii $\lambda$1640 emission line in
J1009$+$26 is its blueshift. As shown in Fig. 11, $v_{50}$ of He ii
$\lambda$1640 is $-$70$\pm{5}$ km s-1 with respect to the systemic velocity
based on the stellar kinematics measured in L20. As a comparison, $v_{50}$ of
O iv] $\lambda$1404, He ii $\lambda$4686, C iv $\lambda$$\lambda$1548, 1550,
and [O iii] $\lambda$5007 are $-$65$\pm{5}$ km s-1, $-$35$\pm{5}$ km s-1,
$-$30$\pm{15}$ km s-1, and $-$20$\pm{15}$ km s-1, respectively. The ionization
potentials (IP) for the corresponding ions are 54.9 eV, 54.4 eV, 47.9 eV, 35.1
eV, respectively. O iv] and He ii $\lambda$1640 emission lines are the most
blueshifted ones among these lines.

One possible origin of the blueshifted He ii $\lambda$1640 emission line is a
highly-ionized AGN wind. As mentioned in Section 4.1, He ii $\lambda$1640 is
likely originating from the highly ionized gas in the vicinity of the AGN. O
iv] lines likely share a similar origin to He ii $\lambda$1640 based on their
similar IP. As a comparison, C iv and [O iii] may have a significant
contribution from the ISM further away from the AGN given their lower IP. This
trend of increasing outflow velocity in higher ionization species in our
objects is also suggested by Bohn et al. (2021) where the outflow velocity
measured from the [Si vi] 1.9630 $\mu$m (IP$\simeq$166.8 eV) is faster than
that from the [O iii] line. These results are all consistent with a
ionization-stratified outflow model for AGN-driven outflow in more luminous
AGN (e.g. Veilleux, 1991; Spoon & Holt, 2009), namely the outflow velocity
decreases with distance to the central ionizing source and so does the
ionization. A more detailed discussion on this combining all multi-wavelength
data is beyond the scope of this paper and thus saved for a future work.

Therefore, the blueshift of He ii $\lambda$1640 and O iv] lines are naturally
caused by the motion of the highly-ionized AGN wind towards the observer,
whereas the blueshift of C iv and [O iii] are smaller due to the contamination
from the nebular emission of the relatively more kinematically quiescent ISM.

Moreover, the blueshift of He ii $\lambda$1640 with respect to He ii
$\lambda$4686 might be caused by the heavier attenuation in the FUV than in
the optical: The red emission line wing of He ii $\lambda$1640 originating
from the material on the far-side of the AGN is attenuated by gas/dust more
severely than that of He ii $\lambda$4686\. The He ii $\lambda$1640 is thus
more blueshifted overall due to the much weaker red emission line wing.
Likewise, the velocity offset between the He ii $\lambda$1640 and [O iii]
emission lines may also partially be caused by this effect, in addition to the
contamination from the ISM in the latter as mentioned above.

##  5 Summary

In this paper, we present the results from a pilot HST/COS spectroscopy
program to examine three dwarf galaxies with primarily AGN-driven outflows
from the sample studied in Liu et al. (2020). The main results are summarized
as follows:

  * •

Blueshifted absorption features are clearly detected in two objects
(J0906$+$56 and J0954$+$47) and tentatively detected in one object
(J1009$+$26), tracing outflows in them.

  * •

In J0954$+$47, fast outflow is detected in multiple transitions including C
ii, C iv, Si ii, and Si iv. The outflow velocity is $\sim-$460 km s-1 for the
high velocity component or $\sim-$230 km s-1 averaged over the entire
absorption feature. In addition, much broader absorption features ($\sim$4000
km s-1) are also seen for C iv and Si iv, which may have a similar origin to
the broad absorption lines in quasars.

  * •

In J0954$+$47, the radius of the outflow, based on the results from
photoionization modeling with CLOUDY, is estimated to be $\sim$0.5 kpc. The
velocity of this outflow exceeds the escape velocity of the system, suggesting
that a large fraction of the outflowing gas may escape the system. This
outflow may help deposit energy into the circumgalactic medium and enrich it
with metals.

  * •

We compare the outflow properties of J0954$+$47 to those of star formation
driven outflows from Chisholm et al. (2015) and Xu et al. (2022): At a given
SFR, this outflow is much faster and possesses a higher kinetic energy outflow
rate than star-forming galaxies. The mass loading factors and momentum outflow
rates are comparable between the two. This is consistent with the scenario
that at a given SFR, compared to stellar feedback, the AGN feedback deposits
more energy to heat the ISM and thus prevents the cooling of gas while having
a similar impact on expelling the gas out. At a given circular
velocity/stellar mass, this outflow possesses a velocity comparable to those
fastest outflows in star-forming galaxies and shows a mass loading factor
close to the highest values seen in the latter. However, the momentum and
kinetic energy outflow rates of this outflow are close to the lowest values
observed in the star-forming galaxies, which is likely caused by the 1–2 dex
higher SFR in sources from Xu et al. (2022) than that in J0954$+$47\. These
results are all based on one dwarf galaxy with AGN, J0954$+$47, and a larger
sample is needed before any robust conclusion can be drawn.

  * •

The AGN in J0954$+$47 is energetic enough to easily drive the fast and
powerful outflow in J0954$+$47\. Nevertheless, we cannot formally rule out the
possibility that star formation activity may also launch the outflow, assuming
the momentum and energy provided by the stellar processes are transferred to
the outflow at high efficiencies ($\gtrsim$60%–230% for momentum and
$\gtrsim$30%–120% for kinetic energy), larger than typically observed in
z$\sim$0 star-forming dwarf galaxies.

  * •

The outflow in J0954$+$47 carries a significant amount of mass, momentum, and
kinetic energy. With a mass loading factor (i.e., mass outflow rate over SFR)
of $\gtrsim$3.4 ($\gtrsim$1.8) averaged over the entire outflow (for the
high-v outflow), the outflow can likely transport more material out of the
galaxy than that ends up in stars. The kinetic energy outflow rate to AGN
luminosity, $\dot{E}_{out}/L_{AGN}$, is $\sim$0.6% ($\sim$1.2%) averaged over
the entire outflow (for the high-v outflow) and is comparable to (larger than)
the expectation from one representative simulation of AGN feedback.
Additionally, this $\dot{E}_{out}$ is $\sim$10$\times$ higher than that
derived from the [O iii] emission line from Liu et al. (2020).

  * •

A strong He ii $\lambda$1640 emission line is detected in both J0954$+$47 and
J1009$+$26 where the transition is covered by the observation. The EW and FWHM
of the He ii, and the C iv/He ii ratios are consistent with an AGN origin.
Such dwarf AGN, if present in abundance in the early universe, may contribute
to cosmic reionization.

  * •

In one object, J1009$+$26, the He ii $\lambda$1640 and O iv] $\lambda$1404
emission lines are the most blueshifted lines among all strong FUV and optical
emission lines with respect to the systemic velocity in this object. These two
emission lines likely trace a highly-ionized AGN wind in this object.

##  Appendix A AGN Luminosity and SFR of J0954$+$47

The AGN luminosity and SFR of J0954$+$47 are the two fundamental physical
properties on which the discussions on the energy source and impact of the
outflow are based. The AGN luminosity can also be obtained by integrating the
AGN spectrum (scaled by the observed monochromatic continuum luminosity at
1320 Å) adopted for the photoionization modeling in Section 3.2.1, which gives
log(LAGN) $\simeq$ 42.8 erg s-1. This is an order of magnitude smaller than
the [O iii] $\lambda$5007 based LAGN. Each of the two estimates of LAGN has
its own caveat. For the FUV-based one, zero extinction is assumed and
log(LAGN) may be underestimated. The departure of the intrinsic AGN radiation
from the adopted AGN spectrum is also not known. For the [O iii]-based one,
the uncertainty of the scaling factor is mainly due to the large scatter of
the scaling relation Lamastra et al. (2009). To make our analysis more self-
consistent, we adopt the FUV-based LAGN in our following discussion on the
outflow detected through the FUV absorption features.

As for the SFR, the [O ii]-based upper limit ($<$0.3 $M_{\sun}\ yr^{-1}$), as
reported in Liu et al. (2020), is obtained by adopting the empirical relation
calibrated for AGN only Ho (2005). In addition, two more upper limits, based
on the FUV luminosity ($<$0.08 $M_{\sun}\ yr^{-1}$) and FUV$+$MIR luminosities
($<$1.9 $M_{\sun}\ yr^{-1}$), are reported in Latimer et al. (2021). Both
values are deemed upper limits as the scaling relations adopted are calibrated
based on star-forming galaxies and the AGN contamination to the FUV and MIR
luminosities are not subtracted. The FUV-only one may be a bit low as it does
not account for the dust-obscured star formation activity. The FUV$+$MIR-based
one is too high: J0954$+$47 is classified as AGN in all popular WISE-based MIR
diagnosis (e.g. Jarrett et al., 2011; Mateos et al., 2012)888As far as we are
concerned, the only exception is the diagnosis proposed by Assef et al. (2010)
adopting all 4 WISE band.. The MIR luminosity is thus likely dominated by the
AGN rather than the star formation, leading to a significant overestimate of
SFR. In all, we adopt the [O ii]-based upper limit on SFR in the following
discussions.

Figure 12: Full G160M spectrum for J0906$+$56\. The expected locations of
strong transitions at the systemic velocity are marked in red dashed lines.
The expected locations of strong z$=$0 foreground absorption features are
marked in blue dashed lines. Figure 13: Same as Fig. 12 but for J0954$+$47.
Figure 14: Same as Fig. 12 but for J1009$+$26.

##  Appendix B Full G160M Spectra of Our Objects

The full HST/COS G160M spectra of J0906$+$56, J0954$+$47 and J1009$+$26 are
shown in Fig. 12, 13 and 14.

We thank the anonymous referee for constructive comments that have improved
the paper. W.L. and S.V. acknowledge partial support for this work provided by
NASA through grant numbers HST GO-15662.001-A, GO-15662.001-B and 15915.002-A.
T.M.T. acknowledges partial support for this work provided by NASA through
grant numbers HST GO-15915.004-A from the Space Telescope Science Institute,
which is operated by AURA, Inc., under NASA contract NAS 5-26555. Based on
observations made with the NASA/ESA Hubble Space Telescope, and obtained from
the Hubble Legacy Archive, which is a collaboration between the Space
Telescope Science Institute (STScI/NASA), the Space Telescope European
Coordinating Facility (STECF/ESA) and the Canadian Astronomy Data Centre
(CADC/NRC/CSA). This research has made use of the NASA/IPAC Extragalactic
Database (NED), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration.

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# Hyperspaces of dimension 1

Alfredo Zaragoza  Departamento de Matemáticas, Facultad de Ciencias,
Universidad Nacional Autónoma de México, Circuito Exterior s/n, Ciudad
Universitaria, Coyoacán, 04510, Mexico city, Mexico<EMAIL_ADDRESS>
###### Abstract.

In a previuos paper the author asked if there exists a one-dimensional space
$X$ that is not almost zero-dimensional, such that the dimension of the
hyperspace of compact subsets of $X$ is one-dimensional. In this short note we
give examples of spaces $X$ that are not almost zero-dimensional such that $X$
is one-dimensional and their hyperspace of compacta of $X$ also is one-
dimensional.

###### Key words and phrases:

Erdős space, almost zero-dimensional space, cohesive space, Vietoris
hyperspace, one-dimensional.

###### 2010 Mathematics Subject Classification:

Primary: 54F65, Secondary: 54F50, 54A10, 54B20, 54H05.

This work is part of the doctoral work of the author at UNAM, Mexico city,
under the direction of the Hernández-Gutiérrez. This research was supported by
a CONACyT doctoral scholarship with number 696239.

##  1\. Introduction

All spaces will be assumed to be separable and metrizable. A space $X$ is
zero-dimensional if it has a base of clopen sets. A space $X$ is one-
dimensional if and only if it has a base $\beta$ of neighborhoods such that
$bd_{X}(U)$ and is zero-dimensional and nonempty for any $U\in\beta$. If $X$
has dimension one we write $dim(X)=1$. In general we can define the dimension
of a space $X$ for any $n\in\mathbb{N}$ (see [7]) but in this work we will
only use the definition of dimension 0 and 1. For a space $X$,
$\mathcal{K}(X)$ denotes the hyperspace of non-empty compact subsets of $X$
with the Vietoris topology; for any $n\in\mathbb{N}$, $\mathcal{F}_{n}(X)$ is
the subspace of $\mathcal{K}(X)$ consisting of all the non-empty subsets that
have cardinality less or equal to $n$; and $\mathcal{F}(X)$ is the subspace of
$\mathcal{K}(X)$ of finite subsets of $X$. For $n\in\mathbb{N}$ and subsets
$U_{1},\ldots,U_{n}$ of a topological space $X$, we denote by $\langle
U_{1},\ldots,U_{n}\rangle$ the collection
$\left\\{F\in\mathcal{K}(X):F\subset\bigcup_{k=1}^{n}U_{k},F\cap
U_{k}\neq\emptyset\textit{ for }k\leq n\right\\}$. Recall that the Vietoris
topology on $\mathcal{K}(X)$ has as its canonical base all the sets of the
form $\langle U_{1},\ldots,U_{n}\rangle$ where $U_{k}$ is a non-empty open
subset of $X$ for each $k\leq n$. Note that if $X$ is a separable metrizable
space, then they every subspace of $\mathcal{K}(X)$ is also a separable
metrizable space (see [5, Theorem 3.3 and Propositions 4.4 and 4.5.2]). In [9]
it was shown that if $X$ is a almost-zero dimensional space, then
$dim(X)=dim(\mathcal{K}(X))$. We are going to show spaces $X$ of one dimension
that are not almost zero dimensional such that $dim(\mathcal{K}(X))=1$. The
main results of this work are:

###### 1.1. Theorem

There exists a connected space $X$ such that $dim(X)=dim(\mathcal{K}(X))=1$.

###### 1.2. Theorem

There exists a totally disconnected space $X$ which is not AZD such that
$dim(X)=dim(\mathcal{K}(X))=1$.

The Theorem 1.1 was suggested in a letter by Roman Pol.

##  2\. Preliminaries

A space $(X,\mathcal{T})$ is almost zero-dimensional(AZD) if there is a zero-
dimensional topology $\mathcal{W}$ in $X$ such that $\mathcal{W}$ is coarser
than $\mathcal{T}$ and has the property that every point in $X$ has a local
neighborhood base consisting of sets that are closed with respect to
$\mathcal{W}$. This concept was introduced by Oversteegen and Tymchatyn in
[3]. They proved that almost zero-dimensional spaces are at most
1-dimensional. Recall that Erdős space is defined as:

$\mathfrak{E}=\\{(x_{n})_{n\in\omega}\in\ell^{2}:x_{i}\in\mathbb{Q},\textit{for
all }i\in\omega\\};$

and complete Erdős space as

$\mathfrak{E}_{\mathrm{c}}=\\{(x_{n})_{n\in\omega}\in\ell^{2}:x_{i}\in\\{0\\}\cup\\{1/n:n\in\mathbb{N}\\}\textit{
for all }i\in\omega\\}$

when $\ell^{2}$ is the Hilbert space of all square summable real sequences.
It’s known that Erdős space, and complete Erdős space are almost zero-
dimensional spaces which are not zero-dimensional spaces(see [1]). A space $X$
is called cohesive if every point of the space has a neighborhood that does
not contain nonempty proper clopen subsets of $X$.

###### 2.1. Lemma[2]

$\mathfrak{E}$ and $\mathfrak{E}_{c}$ are cohesive spaces.

A one-point connectification of a space $X$ is a connected extension $Y$ of
the space such that the remainder $Y\setminus X$ is a singleton.

###### 2.2. Example

Let $p$ be a point outside $\mathfrak{E}_{c}$, consider
$\mathfrak{E}_{c}^{+}=\mathfrak{E}_{c}\cup\\{p\\}$ whose neighbourhoods of
$\\{p\\}$ are the complements of closed bounded sets of $\mathfrak{E}_{c}$.
Then $\mathfrak{E}_{c}^{+}$ is metric separable connected space.

It is known that if a space admits a one-point connectification, then it is
cohesive. Moreover if an almost zero-dimensional space is cohesive, then it
admits a one point connectification (see [1, Proposition 5.4, p. 22]).

Let $X$ be an $AZD$ and cohesive space (for example $\mathfrak{E}$,
$\mathfrak{E}_{c}$), then $X$ has a one-point connectification. Suppose that
$Y=\\{p\\}\cup X$ where $p\notin X$. Since $Y$ is connected then $Y$ is not an
AZD space.

Now let us consider $\mathfrak{E}_{c}^{+}$ of example 2.2 and
$N=\\{0\\}\cup\\{1/n:n\in\mathbb{N}\\}$. Let

$P=[\mathfrak{E}_{c}\times\\{1/n:n\in\mathbb{N}\\}]\cup(p,0)$

with the topology inherited from $\mathfrak{E}_{c}^{+}\times N$, then is a
totally disconnected and is not an AZD space (see [4, Example 3.6]). The
spaces $Y$ and $P$ are the spaces we will use to prove the main result.

Another important result for proving the main Theorems is the following:

###### 2.3. Proposition

[9, Proposition 2.2] $X$ is an $AZD$ space if only if $\mathcal{K}(X)$ is an
$AZD$ space.

##  3\. Proof of main Theorems

Let $Z\in\\{P,Y\\}$ and $d$ a metric for $Z$. For each $n\in\mathbb{N}$, let
$B_{n}=\\{z\in Z:d(z,q)<1/n\\}$ and let $E_{n}=Z\setminus B_{n}$, where $q=p$
if $Z=Y$ or $q=(p,0)$ if $Z=P$. Note that for any $n\in\mathbb{N}$, $E_{n}$ is
an $AZD$ space and, by Proposition 2.3, $\mathcal{K}(E_{n})$ is an $AZD$
space. Let
$\mathcal{N}=\prod_{n\in\mathbb{N}}[\mathcal{K}(E_{n})\cup\\{\emptyset\\}]$,
then $\mathcal{N}$ is an $AZD$ space (let’s consider the set $\\{\emptyset\\}$
as an isolated point of $\mathcal{K}(E_{n})\cup\\{\emptyset\\}$).

Let

$\mathcal{L}=\\{(K_{1},K_{2},\ldots)\in\mathcal{N}:\textit{for }m\geq
n,K_{m}\cap E_{n}=K_{n}\\},and$ $\mathcal{S}=\\{H\in\mathcal{K}(Z):q\in H\\}$

Let’s consider the following functions $\mathcal{G}:\mathcal{L}\to\mathcal{S}$
and $\mathcal{G}_{n}:\mathcal{S}\to\pi_{n}[\mathcal{L}]$ (where $\pi_{n}$ is
the projection to the $n$-th coordinate) such that

$\mathcal{G}(K_{1},K_{2},\ldots)=\\{q\\}\cup\bigcup_{n\in\mathbb{N}}K_{n},and$
$\mathcal{G}_{n}(K)=K\cap E_{n}$

###### 3.1. Lemma

$\mathcal{G}$ is well defined and is a homeomorphism.

###### Proof.

To prove that $\mathcal{G}$ is well defined, let $\mathcal{U}$ be an open
cover of $\\{q\\}\cup\bigcup_{n\in\mathbb{N}}K_{n}$ in $Z$. Since
$q\in\\{q\\}\cup\bigcup_{n\in\omega}K_{n}$, we can suppose that
$B_{m}\in\mathcal{U}$ for some $m$. Note that
$(\\{q\\}\cup\bigcup_{n\in\mathbb{N}}K_{n})\setminus K_{m}\subset B_{m}$,
since $Z\setminus K_{m}\subset B_{m}$. As $\\{U\cap E_{m}:U\in\mathcal{U}\\}$
is an open cover of $K_{m}$ and $K_{m}$ is a compact subset of $E_{m}$, then
there exists $U_{1},\ldots,U_{k}$, such that $K_{m}\subset\bigcup_{i\leq
k}(U_{i}\cap E_{m})$. Therefore $\\{B_{m},U_{1},\ldots,U_{k}\\}$ is a finte
subcover of $\mathcal{U}$, that is $\\{q\\}\cup\bigcup_{n\in\mathbb{N}}K_{n}$
is a compact subset of $Z$. Therefore $\mathcal{G}$ is well defined.

Let’s prove that $\mathcal{G}$ is injective, let
$\widehat{K}=(K_{1},K_{2},\ldots),\widehat{H}=(H_{1},H_{2},\ldots)\in\mathcal{L}$
such that $\widehat{H}\neq\widehat{K}$, then there exists $k\in\mathbb{N}$ so
that $H_{k}\neq K_{k}$. Therefore there exists $x\in H_{k}\setminus K_{k}$,
then $x\in\mathcal{G}(\widehat{H})$ and $x\notin\mathcal{G}(\widehat{K})$.
That is, $\mathcal{G}$ is injective. Now let’s see that $\mathcal{G}$ is
surjective, let $K\in\mathcal{S}$. We define $K_{n}=K\cap E_{n}$, since
$E_{n}$ is a closed subset in $Z$, then $K_{n}$ is empty or is a compact
subset of $E_{n}$. Then $K_{n}\in\mathcal{K}(E_{n})\cup\\{\emptyset\\}$ for
each $n\in\mathbb{N}$, therefore $(K\cap E_{1},\ldots)\in\mathcal{L}$ and
$\mathcal{G}((K\cap E_{1},\ldots))=K$. That is, $\mathcal{G}$ is surjective.
Before proving that $\mathcal{G}$ is a homeomorphism, let’s show that

$\mathcal{B}=\\{\langle
U_{1},\ldots,U_{n},B_{k}\rangle\cap\mathcal{S}:n,k\in\mathbb{N}\textit{ and
}U_{1},\ldots,U_{n}\textit{ are }$ $\textit{open subsets of
}Z\setminus\\{p\\}\\}\cup\\{\langle B_{k}\rangle:k\in\mathbb{N}\\}$

is a basis for $\mathcal{S}$.

Let $K\in\mathcal{S}$ and $\mathcal{W}=\langle W_{1},\ldots,W_{n}\rangle$ an
open subset of $\mathcal{K}(Z)$ such that $K\in\mathcal{W}.$ If
$K_{n}\neq\emptyset$ for some $n\in N$, then $H_{r}\neq\emptyset$ for $r\geq
n$, without loss of generality we can assume that $n=1$. As $p\in
K\in\mathcal{W}$, then there exist $j\leq n$ such that
$p\in\bigcap\\{W_{j}:j\leq n,p\in W_{j}\\}$, and $k\in\mathbb{N}$ such that
$p\in B_{k}\subset\bigcap\\{W_{j}:j\leq n,p\in W_{j}\\}$. To find an element
$\mathcal{V}$ of the base $\mathcal{B}$ such that
$K\in\mathcal{V}\subset\mathcal{W}$, we consider two cases. If $K\setminus
B_{k}=\emptyset$ or if $K\setminus B_{k}\neq\emptyset$. If $K\setminus
B_{k}=\emptyset$ then $K\subset B_{k}$. Therefore $K\in\langle
B_{k}\rangle\subset\mathcal{W}$. If $K\setminus B_{k}\neq\emptyset$ then for
each $x\in K\setminus B_{k}$ there exist $U_{x}$ such that $x\in
U_{x}\subset\bigcap\\{W_{j}:j\leq n,x\in W_{j}\\}$, as $K\setminus B_{k}$ is
compact and $\\{U_{x}:x\in K\setminus B_{k}\\}$ is an open cover of
$K\setminus B_{k}$, there exist $x_{1},\ldots,x_{l}$ such that $K\setminus
B_{k}\in\langle U_{x_{1}},\ldots,U_{x_{l}}\rangle$. Let $\mathcal{V}=\langle
U_{x_{1}},\ldots,U_{x_{l}},B_{k}\rangle\cap\mathcal{S}$, note that
$K\in\mathcal{V}$, and $\mathcal{V}\subset\mathcal{W}\cap\mathcal{S}$. On the
other hand if $K=\\{p\\}$, there exist $k\in\mathbb{N}$ such that $p\in B_{k}$
and $p\in B_{k}\subset\bigcap\\{W_{j}:j\leq n,p\in W_{j}\\}$ this implies that
$K\in\langle B_{k}\rangle\subset\mathcal{W}$. Therefore $\mathcal{B}$ is a
basis for $\mathcal{S}$.

Let $K=(H_{1},\ldots,H_{n},\ldots)\in\mathcal{L}$, and
$\mathcal{U}\in\mathcal{B}$ such that $H=\mathcal{G}(K)\in\mathcal{U}$. If
$H=\\{p\\}$, then $\mathcal{U}=\langle B_{k}\rangle$ for some $k\in\mathbb{N}$
and $H_{n}=\emptyset$ for each $n\in\mathbb{N}$. Let
$W=\\{\emptyset\\}^{k}\times[\langle
B_{k}\rangle\cap(\mathcal{K}(E_{k+1})\cup\\{\emptyset\\})]\times\prod_{m>k+1}[\mathcal{K}(E_{m})\cup\\{\emptyset\\}]$,
note that $K\in W$, and $\mathcal{G}[W]\subset\mathcal{U}$. If
$H_{i}\neq\emptyset$ for some $i\in\mathbb{N}$, then $H_{r}\neq\emptyset$ for
$r\geq n$, without loss of generality we can assume that $i=1$, and that
$\mathcal{U}=\langle U_{1},\ldots,U_{n},B_{k}\rangle$ for some
$n,k\in\mathbb{N}$ . Let $A=\\{j\in\mathbb{N}:U_{l}\cap
H_{j}\neq\emptyset\textit{ for all }l\leq n\\}$ and as
$\\{H_{k}:k\in\mathbb{N}\\}$, is not finite, then $A\neq\emptyset$. Let
$r=\min A$, if $r<k$, then $F_{k}\cap U_{j}=\emptyset$ for some $j\leq n$, so
$F_{r}\cap U_{j}\neq\emptyset$ and $F_{r}\cap U_{j}\subset B_{k}$. Let

$N=\\{(F_{1},F_{2},\ldots)\in\mathcal{L}:F_{r}\in\langle
U_{1},\ldots,U_{n},B_{k}\rangle\\}.$

Note that $K\in N$, if $F=(F_{1},F_{2},\ldots)\in N$ and
$\mathcal{G}(F_{1},F_{2},\ldots)=F$, then $p\in F\setminus F_{k}\subset
B_{r}\subset B_{k}$, so $F\in\mathcal{U}$. If $r\leq k$, then $H_{k}\setminus
H_{r}\subset\bigcup_{j\leq n}U_{j}$ and $H\setminus H_{k}\subset B_{k}$. Let

$N=\\{(F_{1},F_{2},\ldots)\in\mathcal{L}:F_{k}\in\langle
U_{1},\ldots,U_{n}\rangle\\}.$

Note that $K\in N$. If $(F_{1},\ldots F_{k},\ldots)\in N$, and
$\mathcal{G}(F_{1},\ldots F_{k},\ldots)=F$, then $p\in F\setminus F_{k}\subset
B_{k}$, so $F\in\mathcal{U}$. This implies that $\mathcal{G}$ is a continuous
function.
Finally we will show that $\mathcal{G}^{-1}$ is a continuous function, if $U$
is a basic open subset of $\mathcal{L}$, then $U=(\bigcap_{j\in
F}\pi_{j}^{\leftarrow}[W_{j}])\cap\mathcal{L}$, where $W_{j}$ is an open
subset of $\mathcal{K}(E_{j})\cup\\{\emptyset\\}$ and $F$ is a finite subset
of $\mathbb{N}$. Hence

$(\mathcal{G}^{-1})^{\leftarrow}[U]=\bigcap_{j\in
F}(\mathcal{G}^{-1})^{\leftarrow}[\pi_{j}^{\leftarrow}[W_{j}]]\cap\mathcal{S}=\bigcap_{j\in
F}\mathcal{G}_{n}^{\leftarrow}[W_{j}].$

So that is enough to show the continuity of $\mathcal{G}_{n}$ for any $n$. To
prove that $\mathcal{G}_{n}$ is continuous, it is sufficient to show that
$\mathcal{G}_{n}^{\leftarrow}[\langle U_{1},\ldots
U_{k}\rangle\cap\mathcal{K}(E_{n})]$ and
$\mathcal{G}_{n}^{\leftarrow}[\\{\emptyset\\}]$ are open subsets of
$\mathcal{S}$, where $U_{1},\ldots,U_{k}$ are open subsets of
$Z\setminus\\{p\\}$ such that $E_{n}\cap U_{j}\neq\emptyset$ for each $j\leq
k$. We will show that

$\mathcal{G}_{n}^{\leftarrow}[\langle U_{1},\ldots
U_{k}\rangle\cap\mathcal{K}(E_{n})]=\mathcal{S}\cap\langle U_{1},\ldots
U_{k},B_{n}\rangle$

and that

$\mathcal{G}_{n}^{\leftarrow}[\\{\emptyset\\}]=\mathcal{S}\cap\langle
B_{n}\rangle$

Let $H\in\mathcal{S}\cap\langle U_{1},\ldots U_{k},B_{n}\rangle$, then
$H_{n}\neq\emptyset$, $H\setminus H_{n}\subset B_{n}$, and $H_{n}\in[\langle
U_{1},\ldots$ $U_{k}\rangle\cup\\{\emptyset\\}]\cap\mathcal{K}(E_{n})$, thus
$H\in\mathcal{G}_{n}^{\leftarrow}[\langle U_{1},\ldots
U_{k}\rangle\cap\mathcal{K}(E_{n})]$. Let
$F\in\mathcal{G}_{n}^{\leftarrow}[\langle U_{1},\ldots
U_{k}\rangle\cap\mathcal{K}(E_{n})]$, then $F_{n}=F\cap E_{n}\in\langle
U_{1},\ldots U_{k}\rangle\cap\mathcal{K}(E_{n})$ and $F\setminus F_{n}\subset
B_{n}$ then $F\in\langle U_{1}\ldots,U_{k},B_{n}\rangle$. Let
$H\in\mathcal{S}\cap\langle B_{n}\rangle$, then $H_{n}=\emptyset$, therefore,
$H_{n}\in\\{\emptyset\\}$, thus
$H\in\mathcal{G}_{n}^{\leftarrow}[\\{\emptyset\\}]$. Let
$F\in\mathcal{G}_{n}^{\leftarrow}[\\{\emptyset\\}]$, then $F_{n}=F\cap
E_{n}=\emptyset$, thus $F\subset B_{n}$ then $F\in\langle B_{n}\rangle$. This
implies that $\mathcal{G}_{n}$ is a continuous function. Therefore
$\mathcal{G}$ is a homeomorphism.

∎

###### 3.2. Theorem

$dim(Z)=dim(\mathcal{K}(Z))=1$

###### Proof.

Note that $\mathcal{K}(Z)=\mathcal{K}(Z\setminus\\{q\\})\cup\mathcal{S}$. As
$\mathcal{K}(Z\setminus\\{q\\})$ is an AZD cohesive space, then
$dim(\mathcal{K}(Z\setminus\\{q\\}))=1$. By Theorem 3.1, $S$ is homeomorphic
to $\mathcal{L}$ and $dim(\mathcal{L})=1$ because $\mathcal{L}$ is an AZD, but
$\mathcal{L}$ is not zero dimensinal space. This implies that
$dim(\mathcal{S})=1$. Thus $dim(\mathcal{K}(Z))=1$. ∎

Proof of Theorem 1.1

###### Proof.

Let $Y$ be a one-point connectification of $\mathfrak{E}$. By Theorem 3.2 we
have the result. ∎

###### 3.3. Corollary

Let $X$ be an cohesive and AZD spce. If $Y$ is a one-point connectification of
$X$, then $dim(Y)=dim(\mathcal{K}(Y))=1$.

###### Proof.

By Theorem 3.2 we have the result. ∎

Proof of Theorem 1.2

###### Proof.

Consider the space $P$. By Theorem 3.2 we have the result. ∎

Note that the spaces given in the Corollaries 1.1 and 1.2 are unions of $AZD$
spaces. A natural question is:
Does every space $Z$ of dimension 1 that is not $AZD$ and is a finite union of
subspaces $AZD$ satisfy that $dim(\mathcal{K}(Z))=1$ ? The answer to this
question is negative because $[0,1]$ is not an $AZD$ space, but is a union of
$\mathbb{Q}\cap[0,1]$ and $\mathbb{P}\cap[0,1]$ which are $AZD$ spaces, and
$dim(\mathcal{K}([0,1]))$ is not 1.

On the other hand it is known that if $X$ is a compact space of dimension 1,
then the dimension of $\mathcal{K}(X)$ is not finite (see [8, pag 123]). This
implies that if a space $X$ has a compact subset of dimension 1 then the
dimension of $\mathcal{K}(X)$ is not finite. Then for $\mathcal{K}(X)$ to have
dimension 1 each $A\in\mathcal{K}(X)$ must have dimension zero. With the
following Theorem, we will show that it is not enough that the compact subsets
of a space $X$ of dimension 1 have dimension 0 for that hyperspace of compact
subsets of $X$ have to dimension 1.

###### 3.4. Theorem

[6, Theorem 4.1] There exists a space $X$ of dimension 1 such that all its
compacta have dimension 0 and $dim(X^{2})=2$.

###### 3.5. Example

Let $Y=X\times\\{0,1\\}$ where $X$ is as in Theorem 3.4, then $dim(Y)=1$ and
for each compact subset $F$ of $X$ we have that $dim(F)=0$. Let
$f:X^{2}\to\mathcal{K}(X)$ given by $f(x,y)=\\{(x,0),(y,1)\\}$. Note that $f$
is an embedding. This implies that $dim(\mathcal{K}(Y))\geq 2$.

###### 3.6. Question

Let $X$ be a space of dimension 1, such that $dim(X^{\omega})=1$ and for each
$A\in\mathcal{K}(X)$, $dim(A)=0$. Does $\mathcal{K}(X)$ have dimension 1?.

##  4\. Acknowledgement

I would like to to thank Roman Pol for useful suggestions concerning the topic
of this paper.

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