# The Sun’s Magnetic Power Spectra over Two Solar Cycles. II. Cycle Dependence
of Active Region, Magnetic Network, and Their Relation

Yukun Luo  School of Space and Environment, Beihang University, Beijing,
People’s Republic of China  Key Laboratory of Space Environment Monitoring and
Information Processing of MIIT, Beijing, People’s Republic of China  Jie Jiang
School of Space and Environment, Beihang University, Beijing, People’s
Republic of China  Key Laboratory of Space Environment Monitoring and
Information Processing of MIIT, Beijing, People’s Republic of China  Ruihui
Wang  School of Space and Environment, Beihang University, Beijing, People’s
Republic of China  Key Laboratory of Space Environment Monitoring and
Information Processing of MIIT, Beijing, People’s Republic of China

###### Abstract

The multi-scaled solar magnetic field consists of two major components: active
regions (ARs) and magnetic network. Unraveling the cycle-dependent properties
and interrelations of these components is crucial for understanding the
evolution of the solar magnetic field. In this study, we investigate these
components using magnetic power spectra derived from high-resolution and
continuous synoptic magnetograms since cycle 23 onwards. Our results show that
the size of the magnetic network ranges from 26 Mm to 41 Mm without dependence
on the solar cycle. The power of the network field ($P_{NW}$) accounts for
approximately 20% of the total power during any phase of solar cycles. In
contrast to the AR power ($P_{AR}$), $P_{NW}$ displays a weaker cycle
dependence, as described by the relationship $P_{NW}$ $\approx$ 0.6* $P_{AR}$
\+ 40. The power-law index between AR sizes and magnetic network sizes
presents a strong anti-correlation with the activity level. Additionally, our
study indicates that in the absence of sunspots on the solar disc, the
magnetic power spectra remain time-independent, consistently exhibiting
similarity in both shape and power. This study introduces a new method to
investigate the properties of the magnetic network and provides magnetic power
spectra for high-resolution simulations of the solar magnetic field at the
surface at various phases of solar cycles.

magnetic power spectrum - active region - network - cycle dependence

revtex4-1.clsrevtex4-2.cls

##  1 Introduction

Solar magnetic fields, as the main source of solar activity, include two
prominent components: active regions (ARs) and magnetic network (Solanki et
al., 2006). As typical magnetic structures on the solar surface, they also
interact with various scale flow fields like differential rotation and
supergranulation flows (Rincon & Rieutord, 2018). The evolution of ARs may
also influence the distribution and formation of network fields. The
investigation of their interaction and temporal variation with the solar cycle
can help to understand the evolution of magnetic fields, dynamic processes
with flow fields, and their impact on the solar atmosphere (Bellot Rubio &
Orozco Suárez, 2019; van Driel-Gesztelyi & Green, 2015).

As the most obvious magnetic configurations on the solar surface, ARs exhibit
substantial scales and strong enough magnetic fields to yield pronounced,
easily detectable signals (van Driel-Gesztelyi & Green, 2015). Decades of
continuous observations of the solar magnetic fields have resulted in a wealth
of data on ARs. These lead to the well-studied cycle dependence of ARs.
However, we know little about magnetic networks due to their small spatial
scales and relatively rapid evolution (Bellot Rubio & Orozco Suárez, 2019).
Consequently, there is an ongoing debate about the properties of the magnetic
network.

The first ongoing debated question is the origin of the magnetic network.
According to Martin (1990), ephemeral regions (ERs) contribute 90$\%$ or more
of the network flux, with the remaining flux originating from the internetwork
fields. Schrijver et al. (1997) build upon this conjecture and construct a
model to investigate the evolution of the network flux, ignoring the
contribution of the magnetic internetwork. Their findings support ERs as the
main source of the network, with only a small amount of flux dispersing from
ARs. Hagenaar et al. (2003) conduct further investigation and suggest the
dominance of ERs in the relatively small network, while the larger scale
network results from decaying ARs. However, Gošić et al. (2014) argue that the
internetwork is the most important source, not ERs. As these efforts present
different potential sources of the network flux, further investigation is
needed.

The second open question is whether the network flux and size vary with the
solar cycle. Answering this question is not only necessary to understand the
formation and sustainment of the network but also to shed light on the process
of flux diffusivity from large concentrations to the network (Leighton, 1964).
Hagenaar et al. (2003) suggest that the network with fluxes $\leq 2\times
10^{19}$ Mx is independent of solar cycles, while the stronger network
increases with solar activities because of the antiphase relationship between
ERs and ARs. However, we still know little about the variation of total
network flux (Meunier, 2003).

Previous attempts to determine the variation of network size with the solar
cycle use different proxies, with divergent results. Hagenaar et al. (1997),
focusing on the chromospheric network, propose no dependence of network sizes
on local magnetic strength. Both regarding the chromospheric network, Singh &
Bappu (1981); Berrilli et al. (1999) find a tendency toward smaller network
sizes at the solar maximum, while Wang (1988); Muenzer et al. (1989) report a
contrary trend. Focusing on supergranulation, De Rosa & Toomre (2004) conduct
studies direct from Dopplergram and support the anti-correlated cycle
dependence of supergranulation sizes. Meunier et al. (2007a) use intensity
maps to study the variation of the supergranulation size with different parts
of magnetic fields. They find that larger supergranulation is associated with
stronger network fields, but sizes decrease with increasing magnetic strength
within supergranulation. In addition to observational approaches, simulation-
based studies by Crouch et al. (2007); Thibault et al. (2012, 2014) employ the
diffusion-limited aggregation model to support the positive dependence of
characteristic network sizes on solar activity.

However, proxies on magnetic network studies may introduce unexpected
discrepancies. For example, the dynamical interaction between supergranulation
and magnetic network is not well understood (Rincon & Rieutord, 2018). The
magnetic network has a large uncertainty range in relation to the
chromospheric network (Schrijver et al., 1996). Investigating the magnetic
network based on magnetograms can directly answer whether the network
properties vary with the solar cycle. The power spectrum obtained from
magnetograms is a useful tool to identify and study magnetic structures and
can be used to measure the typical scale at which the magnetic fields are
organized. Magnetic power spectra obtained by Abramenko et al. (2001) reveal
suspected network structures. In the first paper of the series, Luo et al.
(2023) (hereafter referred to as Paper I) also identify magnetic structures at
supergranulation scales within the power spectra. Hence, we can expect to
identify the network in magnetic power spectra and investigate network
properties and origin based on the power spectra features related to the
network, such as scale and power-law index.

As the second paper in the series, we improve our identification methods for
determining AR sizes and network sizes from magnetic power spectra. The
analysis is extended to cover the entire solar cycles 23, 24, and part of
cycle 25. We use Solar Dynamics Observatory (SDO)/Helioseismic and Magnetic
Imager (HMI) and Solar and Heliospheric Observatory (SOHO)/Michelson Doppler
Imager (MDI) synoptic magnetograms. The calibration method proposed by Paper I
is applied to ensure comparable and homogeneous identification results. Our
goal is to examine how the size and power of the network vary with solar
cycles, the relation between ARs and the network, and to attempt to reveal the
possible origin of the magnetic network. Additionally, the spotless days
during the three solar minimums provide an opportunity to speculate on the
network properties during grand minima.

This paper is organized as follows. In Section 2, we describe the HMI and MDI
synoptic magnetograms and improve the methods for determining AR sizes and
network sizes. Section 3 subsequently presents the identification results and
explains variations of the magnetic network with solar cycles. The relation
between AR power and network power is also presented in this section. We
summarize and discuss our results in Section 4.

##  2 Data and Methods

###  2.1 Data

This paper uses radial synoptic magnetograms observed by MDI on board SOHO
(Scherrer et al., 1995) and HMI on board SDO (Scherrer et al., 2012; Schou et
al., 2012), respectively. They cover cycles 23, 24, and part of cycle 25,
beginning in Carrington Rotation (CR) 1911 (1996 July) and ending in CR 2265
(2022 December). The data sets include a total of 355 CRs.

Same as Paper I, we utilize the ‘pyshtools’ package in Python to perform
spherical harmonic decomposition for synoptic maps (Wieczorek & Meschede,
2018; Wieczorek et al., 2019), subsequently deriving their magnetic power
spectra. To meet the requirements of the algorithm in the package, the grid
size must be $n\times n$ or $n\times 2n$. Therefore, we transform the data
resolution of MDI from 1080$\times$3600 to 1080$\times$2160, resulting in the
maximum spherical harmonic degree $l_{max}$ is 539. Similarly, for HMI data,
the resolution is transformed from 1440$\times$3600 to 1440$\times$2880 and
$l_{max}=719$. To ensure a consistent analysis, we calibrate all HMI magnetic
power spectra by the method proposed in Paper I. Our analysis focuses only on
MDI and HMI power spectra between the range $l=6$ and 539.

###  2.2 Determining AR Sizes and Network Sizes

Figure 1: Examples of the four types of magnetic power spectra (left) and the
corresponding synoptic maps (right). From top to bottom are CRs 2063, 1960,
2014, and 2048, respectively. The saturation values of synoptic maps are all
150 G.

Identified AR sizes and network sizes are marked by the vertical gray dashed
line. The red dashed lines are the fitting ones of power spectra between AR
sizes and network sizes.

Based on the relative strength between AR power and network power, there are
four types of magnetic power spectra. Figure 1 shows the typical examples for
each type. The typical supergranular size is generally regarded as
approximately 36 Mm (Williams et al., 2014; Hathaway et al., 2015; Rincon &
Rieutord, 2018), so we restrict the identification range to be $l=105\sim 170$
(26 Mm $\sim$ 42 Mm). This aims to minimize the effects caused by similar
scale structures like ERs. We’ll discuss the effect of different
identification ranges in Section 3.2.1. Below are the four types of magnetic
power spectra, based on which we identify the typical network size using
different methods.

The top panel of Figure 1 shows the first type. During the solar minimum,
there are no AR features, only a peak corresponds to the network. We identify
the positions of the peaks through an available peak-finding algorithm:
‘scipy.signal.find$\\_$peaks’ within Python (Virtanen et al., 2020). If there
is only one local maximum that meets the prominence and width threshold within
the identification range, it is the peak we look for. Using this algorithm, we
identify network sizes for 34 CRs.

During the active phase of the solar cycle, the AR power is much stronger than
the network power. As a result, the network appears as the knee, as shown in
the second line of Figure 1. For this type, we use an algorithm similar to
Satopaa et al. (2011) to identify the location of the network. This algorithm
detects the knee by finding the point with the most significant change in
slope, based on the Lagrange median theorem. To do this, one endpoint of the
identification range is fixed, and as the other endpoint moves, the knee is
identified as the point that is parallel to the secant line and appears to be
unmoving. In this type, network sizes for 103 CRs are identified.

If the AR power slightly exceeds the network power, the magnetic network
signal will appear as a small peak instead of a knee. The third line of Figure
1 is an example of this third type. To enhance the peak signal, which is
easily affected by spectral fluctuations, we subtract the baseline of the
power spectrum. We use the asymmetric least squares method in the
‘pybaselines’ package to obtain the baseline (Erb, 2024). We identify the peak
in the processed power spectrum using two morphological methods. The first
method is the same as the one used for the first type. The second method is
developed by Duarte & Watanabe (2021). Their method selects peaks based on
multiple parameters such as the distance and height difference between
neighboring peaks. The peak remains only if it is identified by both methods.
The third type has 117 CRs with identified network sizes.

Some power spectra may not exhibit significant characteristic sizes, or there
may be multiple signals strong enough to be identified within the given range.
For example, in the bottom panel of Figure 1, the AR power has the same
intensity as the network power. There are multiple peaks that may correspond
to ERs or fragments of ARs. We can not distinguish which is the magnetic
network so all peaks are not identified as magnetic network. To ensure the
accuracy of our methods, we exclude this type, i.e., the fourth type of power
spectra from the identification process, resulting in the exclusion of 101
CRs.

For ARs, we restrict the identification range for $l=10\sim 60$ (73 Mm $\sim$
438 Mm) to reduce possible misidentifications. During the active phase, the
power spectrum shows the AR power as the strongest signal, appearing as a peak
(see Figures 1 (c) and (e) for examples). We use the same algorithm as for the
first type of network identification. The locations of identified ARs are
marked by the left dashed vertical lines in the plots. We have identified ARs
in the power spectra of 196 CRs. If the AR is too weak like Figure 1 (a), or
if there are interference signals as shown in Figure 1 (g), we exclude the
power spectrum from the identification process.

##  3 results

###  3.1 AR Typical Size Detected Based on Power Spectrum

Figure 2: Probability density functions of AR areas obtained by the database
of Wang et al. (2023) (black) and power spectra (red), respectively.

The size of ARs we identified ranges from $l=12$ (365 Mm) to $l=56$ (78 Mm).
Assuming that the identified ARs are both the circular bipolar magnetic region
(BMR), we can convert the size information into areas using the following
equation:

$Area=2\pi{(\frac{1}{4}\frac{{2\pi R_{\odot}}}{l})^{2}},$ (1)

where $R_{\odot}$ is the radius of the Sun and $l$ is the spherical harmonic
degree. The area of ARs in our results ranges from 786 $\mu Hem$ to 17135 $\mu
Hem$. The mean area is 5135 $\mu Hem$ and the corresponding size is about 190
Mm ($l\approx 22$). The probability density function (PDF) of areas is shown
as the red line in Figure 2. The peak of the PDF is located at 3481 $\mu Hem$,
corresponding to 165 Mm ($l\approx 27$), which is slightly smaller than the
mean value.

In previous work, ARs are usually detected morphologically from magnetograms
and then their sizes are measured. Our recent AR database from Wang et al.
(2023) is a typical example, which is used here for comparison. The
corresponding PDF is shown as the black line in Figure 2. Both PDFs are close
to the log-normal distribution commonly found in previous work about ARs
(Abramenko & Longcope, 2005; Canfield & Russell, 2007). However, the PDF
obtained in this paper has a narrower full width at half maxima, indicating
that ARs identified from power spectra tend to concentrate on a narrow range.
Additionally, the mean area and the peak of the PDF from the database are 2360
$\mu Hem$ ($\approx$135 Mm, $l\approx 32$) and 1584 $\mu Hem$ ($\approx$110Mm,
$l\approx 39$), respectively, which are significantly lower than the values of
the red line.

Two aspects may contribute to the discrepancy in mean areas. ARs have
irregular shapes. The AR scale identified in magnetograms morphologically
corresponds to the total number of pixels of the irregular structure. In
contrast, the scale identified in a power spectrum is indicative of the
maximum distance between the edges of a network. Therefore, the areas obtained
by Equation (1) are systematically larger than those obtained by morphological
methods. The second reason is that the stronger signal in the power spectrum
will mask the weaker one. Based on the positive relationship between AR areas
and flux proposed by Sheeley (1966), the smaller ARs are usually related to
weaker ones. Hence, the relatively small ARs tend to be masked, resulting in
systematically larger identified sizes.

###  3.2 Cycle Phase Dependence of Typical Network Size

####  3.2.1 Identification Based on Individual Synoptic Maps

Figure 3: Network sizes identified in magnetic power spectra during solar
cycles 23, 24, and part of 25. Figure 4: Histogram of network sizes identified
in magnetic power spectra.

We first investigate the cycle phase dependence of typical network size based
on individual synoptic maps. Figures 1 (a), (c), and (e) show 3 typical
examples. The vertical dashed lines on the right highlight the identified
network. The identified network sizes range from $l=106$ ($\approx$26 Mm) to
$l=169$ ($\approx$41 Mm) and their time dependence is shown in Figure 3. The
scale range is broad and nearly homogeneous in any phase of the solar cycle,
implying that the probability of the network emerging at any size within the
scale range is approximately equal. Meanwhile, this emphasizes that the
network is also obvious in the active phase not only during solar minimum.
Additionally, there is no significant variation in the scale range for the
period. This is an indication that the network sizes have no significant cycle
phase dependence or cycle dependence. It is worth noting that there are some
spotless days during three solar minimums, implying that the size of the
magnetic network may also have similar properties during the solar grand
minimum.

Although the results presented above come from two datasets with different
spatial resolutions and magnetogram sensitivities, the analysis in Paper I
shows that the discrepancy in the spherical harmonic degree of the network
size between the HMI and MDI synoptic maps, $\Delta l$, is less than 5 after
the calibration. The discrepancy between the two datasets has a negligible
impact on the identification of AR and network sizes.

Figure 4 shows the histogram of identified network sizes. The mean network
size is 33.41 Mm ($l=131$), with a standard deviation of 4.43 Mm. The skewness
and kurtosis are 0.058 and 1.87, respectively. These values are similar to the
characteristic parameters of a uniform distribution. This suggests that our
result is close to a uniform distribution within the identification range.
However, previous work on supergranulation typically shows a unimodal
distribution, as shown by Simon & Leighton (1964) using the autocorrelation
method. This may be contributed by the same reason mentioned in Section 3.1:
the distribution we obtain pertains to the typical network size, which
contains the strongest network power. In addition to this reason, there are
some slight differences between supergranulation and magnetic network (Rincon
& Rieutord, 2018; Requerey et al., 2015, 2017). Hagenaar et al. (1997) report
that the sizes estimated from magnetograms are typically smaller than
Dopplergrams. Therefore, the network size and the supergranulation size could
present different distributions.

The effects of the various identification ranges of $l$ on the typical network
size and its cycle phase dependence, as mentioned in the beginning of Section
2.2, are presented in the Appendix. If using a broader identification range
like $l=100\sim 300$, the identified results smaller than 25 Mm mainly
concentrate on 15 Mm. ERs might contribute to the high concentration of 15 Mm
because the typical ER size is smaller than about 22 Mm (Tang et al., 1984;
Harvey, 1993). This result indicates that the upper limit $l$ of the
identification range should be set to the size corresponding to approximately
25 Mm. When selecting the upper limit of the identification range from $l=170$
(24.5 Mm) to $l=180$ (25.9 Mm), the identified network sizes exhibit minimal
variation and remain cycle-independent. To minimize the potential effects of
ERs, we choose 24.5 Mm, i.e., $l=170$ as the upper limit. Similarly, the lower
limits of the identification range of $l=100,105$, i.e., 43.8 Mm and 41.75 Mm,
yield comparable outcomes.

Some structures of a similar scale like ERs can introduce random interference
in power spectra, bringing the uncertain range of our identification results.
A weak cycle dependency of network sizes could be masked by the uncertainty.
This bias can be reduced by smoothing and averaging power spectra, which will
be investigated in the next subsection.

####  3.2.2 Identification Based on Averaged Spectra over Maximum and Minimum
Phases

Figure 5: Averaged magnetic power spectra at different phases of solar cycles.
Blue: cycle 23 maximum; red: cycle 24 maximum; black: solar minimum across
three solar cycles without any ARs. The vertical dashed lines mark the network
sizes identified.

The identified results in average power spectra during different phases are
displayed in Figure 5. Data in the active phase are selected using the monthly
smoothed total sunspot number (SSN, version 2.0) as a threshold. We choose
$SSN>100$ for cycle 23 (blue lines) and $SSN>64.5$ for cycle 24 (red lines).
The typical network power is 11.8 $G^{2}$ and 6.7 $G^{2}$, respectively. For
the quiet phase (black lines), we use synoptic maps without any spots as the
representation. To account for the limited data that satisfies the conditions,
we combine data from various solar minimums: CRs 2073, 2074, 2082, 2210, 2222,
2223, and 2227. The potential difference of power spectra between various
solar minima is negligible, which is discussed in Section 3.4.2. The typical
network power only reaches 0.2 $G^{2}$, which is significantly lower than that
during the solar maximum. Section 3.4 will examine the meaning and reason for
this difference in more detail.

The typical network sizes for cycle 23 maximum, cycle 24 maximum, and the
solar minimum are 31.3 Mm, 35.9 Mm, and 29.2 Mm, respectively. However, this
data is not sufficient to support the idea that network size increases with
solar activity. It should be noted that the peak in the average power spectrum
of the solar minimum is relatively flat. This means that the power of magnetic
structures near the peak is also strong enough to be considered as network
features. Hence, we believe that the range of typical network sizes can be
extended and even to include the size during the solar maximum.

Combining the above analysis, the conclusion of Section 3.2.1 can be further
confirmed: the network size is not dependent on the solar cycle. A similar
result is given by Hagenaar et al. (1997), who focused on the chromospheric
network and also suggested no dependence of the network size on the local
magnetic strength. Next, we will discuss the cycle dependence of the network
power and its relation with ARs.

###  3.3 Cycle Phase Dependence of the Power Index for the Range between AR
Size and Network Size

Figure 6: Time evolution of power-law indices (top) between AR sizes and
network sizes and SSN (bottom). Figure 7: Power-law indices versus SSN. The
red line is the fitting line.

In Figures 1 (c) and (e), the power spectra between AR sizes and network sizes
are nearly linear. The linear power spectra suggest that the cascade from AR
power to network power is universal. The AR could be one of the sources of the
network, and the diffusivity could be homogeneous across multiple scales. The
power-law index between AR sizes and network sizes can help investigate their
relation.

To determine the power-law index through the least square fitting, we use the
sizes of ARs and the network from the previous subsections as range endpoints.
During the solar minimum, ARs are rare, so we use $l=30$ as the left endpoint
of the fitting range for this period. To avoid biases caused by the shape of
peaks, we choose $l=l_{NW}-10$ as the right endpoint, where $l_{NW}$ is the
location of the identified network. We judge the goodness of fit using the
residual sum of squares (RSS) as a criterion. We exclude power-law indices
with $RSS>0.55$, leaving 191 CRs with indices ranging from -1.02 to 0.98, as
shown in Figure 6. The comparison between the top panel and the bottom panel
shows a significant anti-correlation between the indices and the solar cycle.
Figure 7 displays the scatter plot between indices and SSN. The power-law
indices have a linear relationship with the logarithm of SSN. Through least
square fit, we obtain the following formulation:

$\begin{array}[]{c}k=(-0.54\pm 0.03)*\lg{SSN}+0.48\pm 0.05,\end{array}$ (2)

where $k$ is the power-law index. This equation provides a way to evaluate the
power-law index with a given SSN.

Equation (2) shows that the solar activity modulates the power-law index:
stronger activity results in a smaller index. During the solar active phase of
cycle 23, the power-law indices are universally smaller than those in cycle
24. The average indices for solar active phases are -0.65 and -0.59,
respectively. This is due to cycle 23 being stronger than cycle 24. However,
all indices are smaller than -1.5, as proposed by Kraichnan (1965) about
magnetohydrodynamics turbulence power spectra. This suggests the presence of
energy injection in network scales.

As mentioned in the Introduction, network features are present in the magnetic
power spectra of Abramenko et al. (2001). The power-law index they obtained
also corresponds to the range between AR sizes and network sizes (see Figure 7
in their paper). For local AR magnetograms, their indices are $-0.71\pm 0.02$,
which is close to the indices shown by the red dashed lines in Figures 1 (c)
and (e). The results for local magnetograms of quiet sun are $0.32$ and
$0.49$, smaller than the value of $0.95$ shown in Figure 1 (a), but within the
range of variation we obtained. This supports our identification and fitting
results.

According to Paper I, the difference in the quality of HMI and MDI
magnetograms has a negligible impact on the power law indices after
calibration. The variation of power-law indices is controlled by both ARs and
magnetic network. In the next section, we will quantitatively examine how
these two structures affect the indices.

###  3.4 Understanding the Cycle Phase Dependence of the Power Law Index

####  3.4.1 Cycle Phase Dependence of Network and AR Powers

Figure 8: Comparison of different parts magnetic power. (a): the evolution of
total power (black), network field power (red), and AR power (blue). (b): the
variation of ratios between AR power and network power with the solar cycle.
(c): similar to (b) but for the ratio between network power and total power.
The data are smoothed using a smoothing window of two years. Figure 9:
Magnetic network power versus AR power during the solar active phase.

We use ${P_{AR}}=\sum\limits_{l=10}^{50}{{P_{l}}}$ as the AR power and
${P_{NW}}=\sum\limits_{l=105}^{180}{{P_{l}}}$ as network power, where $P_{l}$
is the power for spherical harmonic degree $l$. The time evolution of these
powers, as well as that of the total power
${P_{Tot}}=\sum\limits_{l=1}^{539}{{P_{l}}}$, is displayed in Figure 8 (a).
The total power is divided by three for comparison. In general, the AR and
network power vary with the total power, with the former being stronger than
the latter most of the time. Only during the solar minimum is the network
power stronger. The minimum value of network power is 19.5 $G^{2}$, which
occurs during the solar minimum between cycle 24 and cycle 25. The maximum
value is 1054 $G^{2}$. It has increased by 54 times. This amplitude is larger
than expected from previous work (Hagenaar et al., 2003). The AR power has a
minimum value of 6 $G^{2}$ and a maximum value of 1748 $G^{2}$. The increasing
factor can be as high as 288, significantly larger than that of network power.

In Figure 8 (b), a comparison is made between AR power and network power. The
ratio has a significant solar cycle dependency, with the AR power being only
0.3 times the network power during the solar minimum. As the ARs emerge
frequently, the ratio increases rapidly until it reaches 1.6 during both the
solar maximum for cycle 23 and cycle 24. The ratio is dominated by AR power
and gradually saturates when it approaches 1.6.

To investigate the relationship between decaying ARs and magnetic networks, we
analyze data from the active phase above the red line in Figure 8 (b). The
comparison between AR power and network power during this phase is shown in
Figure 9. The relationship is

$P_{NW}=(0.61\pm 0.003)*P_{AR}+(39.66\pm 2.86).$ (3)

The intercept term may represent the sources for network flux that are
independent of solar cycles. We suggest these sources could be ERs or
internetwork fields, which are thought to have little or no variation with
solar cycles. The power from these sources is only about 40 $G^{2}$,
significantly smaller than the network power during the solar maximum. This
might imply that the decaying AR is the dominant contributor to network fluxes
during the solar maximum. Figure 8 (b) shows that during the minima of solar
cycles 23 and 24, the magnetic power in ARs is just about one third of the
power in networks. Previous studies (e.g., Harvey et al., 1975; Schrijver &
Harvey, 1994; Chae et al., 2001; Hagenaar, 2001; Jin & Wang, 2015) indicate
that internetwork fields and ERs bring a huge amount of flux to the solar
surface. Internetwork fields and ERs could also be source of the network field
during the solar minima.

The fixed linear relationship may indicate that cancellation between flux
concentrations dissipates approximately $40\%$ of AR power. The remaining
$60\%$ of power is cascaded into network fields. Hence, network power also
exhibits cycle dependence but is weaker than ARs. The cascade ratio remains
constant between various solar cycle phases. This further implies that the
diffusion process from ARs to network regions is similar through multiple
scales and independent of solar cycles.

Based on Equation (3), the variation of power-law indices could be explained.
During the active phase, the AR power cascades to the network in a fixed
ratio. Meanwhile, the approximate constant power is also injected from ERs or
internetwork fields to magnetic networks. As a result, the power-law index
deviates from -1.5, the value proposed by Kraichnan (1965). The influence of
power injection from small scale structures increases as the AR power weakens,
causing the power-law index to increase and deviate further from -1.5. When
ARs rarely emerge on the surface, ER and internetwork fields dominate the
network, resulting in positive indices.

Figure 8 (c) shows the variation of ratios between network power and total
power. This is another representation of the weak cycle dependence of the
network. The ratio varies slightly, with the largest value being $22.1\%$ and
the smallest being $19.3\%$. Taking account of the small amplitude of ratio
variation, we can estimate that the ratio is approximately $20\%$ during any
phase of solar cycles.

####  3.4.2 Similar Magnetic Power Spectra for Magnetograms without ARs

Figure 10: Magnetic power spectra of the 7 CRs of which synoptic magnetograms
without any ARs: CRs 2073 (blue), 2074 (yellow), 2082 (green), 2210 (red),
2222 (purple), 2223 (brown), and 2227 (pink).

Figure 10 displays the seven magnetic power spectra from various solar
minimums without ARs. Their average power spectrum is presented in Figure 5.
All power spectra for $l>40$ exhibit similar profiles and power, and they
roughly overlap. Their power-law indices are around $0.72\pm 0.08$, slightly
higher than that in the quiet-Sun spectrum proposed by Abramenko et al.
(2001). Similarly, Foukal & Milano (2001) analyse Ca II photographic plates
and suggest that the network area does not vary significantly among the nine
solar minima in the last century. In kinetic power spectra, the peaks
corresponding to supergranulation are also nearly constant (Williams &
Pesnell, 2011). We can speculate that the network has similar properties when
there are no ARs on the solar surface.

Considering that during the grand minima, such as the Maunder minimum (Spoerer
& Maunder, 1890; Usoskin, 2023), ARs rarely appear on the surface, the
magnetic power spectra of the 7 CRs shown in Figure 10 could be applied to the
grand minima. Thus, the magnetic power spectra for spotless days provide an
effective way to perceive the magnetic field, and consequently, the solar
irradiance (Dasi-Espuig et al., 2016), etc. during the grand minima.

##  4 Conclusion and Discussion

This paper presents a new approach to measuring both the AR and magnetic
network sizes, as well as investigating their cycle-dependent properties and
relationships. The results show that the size of ARs ranges from 78 Mm to 365
Mm and their corresponding areas follow a log-normal distribution. The
identified network sizes range from 26 Mm to 41 Mm, which are close to the
supergranulation sizes determined from kinetic power spectra by Williams et
al. (2014); Hathaway et al. (2015). But De Rosa & Toomre (2004) and Meunier et
al. (2007b) obtain different values for the supergranulation size using the
local correlation tracking method, which shows a pronounced dependence on the
smoothing procedure (Rincon & Rieutord, 2018). Our results indicate that the
typical network size has no cycle dependence. During the active phases of
cycles 23 and 24, the network is identified in 87 CRs, suggesting that it is a
significant feature not only in the quiet sun.

We also study the cycle dependence of the network power ($P_{NW}$) and its
relation with AR power ($P_{AR}$). We find that the power-law index between AR
sizes and network sizes displays an anti-correlation with solar cycles. The
ratio between network power and total power is approximately 20$\%$ regardless
of the cycle phase. The network power shows a weaker cycle dependence than the
AR power, their relationship is described by $P_{NW}$ $\approx$ 0.6* $P_{AR}$
\+ 40. Based on this relationship, we propose a possible explanation for the
variation of the power-law index. The two terms on the right side of the
equation might correspond to two different sources of the typical magnetic
network and decaying ARs could be the primary source during the solar maximum.

In addition, we find that in the absence of ARs on the solar surface, the
power spectra are time-independent and exhibit similarity in shape and power.
We propose that the magnetic power spectra might exhibit comparable features
during the grand minima such as the Maunder minimum when ARs rarely appear on
the solar surface. The magnetic power spectra of the synoptic maps without
ARs, as presented in Figure 10, might be applicable to the grand minima,
particularly for the spectra with $l>\sim 20$. Hence, these power spectra
provide a new way to estimate properties of magnetic fields during the grand
minima.

The typical network sizes identified are cycle-independent, but the cycle
dependence of supergranulation size is controversial. Meunier et al. (2007a)
propose that strong magnetic network fields are typically associated with
relatively larger supergranulation. Since supergranulation sizes and magnetic
network sizes could have certain differences as reported by Hagenaar et al.
(1997); Rincon & Rieutord (2018), the supergranulation sizes could increase
slightly with the solar cycle, while the typical network sizes remain cycle
independent.

The network field is one of the important contributions to the variation of
total solar irradiance (TSI), which affects Earth’s climate (Solanki et al.,
2013). The cycle dependence of network power can help to understand how the
network affects TSI. While the network power exhibits a correlation with solar
cycles, its ratio to total power is approximately $20\%$ for any cycle phase.
We believe that this correlation also applies to other solar cycles, helping
to calibrate different TSI composites and providing a new constraint for the
historical TSI reconstruction (Dasi-Espuig et al., 2014, 2016). Additionally,
the power spectra without ARs can be used to estimate the variation of TSI
during grand minima.

This paper gives typical magnetic power spectra during solar maximum and
minimum obtained directly from magnetograms. Comparing these spectra with
kinetic power spectra like Hathaway et al. (2000) and Gizon & Birch (2012)
could reveal the relation between kinetic and magnetic energy across various
scales, from global to network scales (Abramenko et al., 2011). These help for
high-resolution simulation of the global magnetic field at the solar surface
(Hotta et al., 2022). Additionally, the spectral features of ARs and the
network, as well as their cycle dependence, can help to reconstruct high-
resolution magnetograms from low-resolution ones. This will be the future
work.

We are deeply indebted to the anonymous referee for the careful and invaluable
comments, which helped us to improve our manuscript. The research is supported
by the National Natural Science Foundation of China No. 12350004, No.
12173005, and National Key R&D Program of China No. 2022YFF0503800. We would
like to express our gratitude to the teams responsible for the development of
Python toolkits such as ‘pyshtools’ and ‘scipy’. The SDO/HMI data are courtesy
of NASA and the SDO/HMI team. SOHO is a project of international cooperation
between ESA and NASA. The sunspot number data are provided by WDC-SILSO, Royal
Observatory of Belgium, Brussels.

Figure 11 shows the identified network sizes and their histograms using
different identification ranges of $l$ comparing with the range used in
Section 2.2. The top panel corresponds to a broad identification range:
$l=100\sim 300$ (14.6 $\sim$ 43.8 Mm), and we get the network sizes from 15.2
to 43.4 Mm. Panel (a) shows that the distribution of results exceeding 25 Mm
is also nearly uniform, whereas the portion below 25 Mm is not. In panels
(c)-(f), using the various upper limits of $l$=180, 175, the lower limits of
identified network sizes extend to 24.5 Mm and 25.2 Mm, respectively. In
panels (g) and (h), we change the lower limit of $l$ from 105 to 100 and get
the upper limit of the identified network size of 43.4 Mm. In all cases, the
identified network sizes are not dependent on the solar cycle.

Figure 11: Identified network sizes (right) and their histograms (left) using
different identification ranges of $l$. From top to bottom, the ranges of $l$
are 100-300, 105-180, 105-175, and 100-170, respectively.

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physically reasonable. We note, however, that our assumptions that the active
bands are relatively well-isolated from the remote bands likely breaks down
for sufficiently large strain.

Taking the identifications of states in our model at face value and
incorporating all sectors, we arrive at the following possible phase diagram
of $\nu=-3$.

  1. 1.

Quantized anomalous hall order at $\varepsilon_{\mathrm{Gr}}\leq 0.05\%$.

  2. 2.

IKS order at $0.05\%<\varepsilon_{\mathrm{Gr}}<0.2\%$.

  3. 3.

A normal metal at $0.2\%\leq\varepsilon_{\mathrm{Gr}}$.

This is shown in Fig. 1(d). We note that the exact phase boundaries are
expected to change somewhat depending on finite-size effects and other
parameters such as $\kappa$. For instance, using $L_{y}=6$ stabilizes QAH at a
larger strain value (as does decreasing $\kappa$ to approach the chiral
limit). Furthermore, the energy differences between these phases are only
slightly larger than the numerical precision of our Hamiltonian and ground
state — and much larger than the physical uncertainty in our model.
Nevertheless, we expect the basic picture of the three phases to be robust.

Finally, we comment on the phase transitions. As QAH and IKS order lie in
different quantum number sectors, we expect a first-order phase transition
between them. For the $(0,0)$ sector, the ferromagnetic IKS order breaks
symmetries, whereas the normal metal does not. The transition between them may
in principle be second-order (though we do not rule out a first-order
transition). If this is the case, any quantum critical behavior would manifest
in the region where we have found the “mixed” order.

Figure S11: DMRG energies in all four sectors as a function of heterostrain
$\varepsilon_{\mathrm{Gr}}$. Due to computational expense, we use $\chi=8192$
except in the $(1,1)$ which is already well-converged by $\chi=4096$. The
$(1,0)$ sector was not computed beyond $\varepsilon_{\mathrm{Gr}}=0.2\%$, but
is expected to be above $(0,0)$ and $(0,1)$ in energy. Note that the $(0,0)$
is clearly the lowest energy for $\varepsilon_{\mathrm{Gr}}>0.2\%$. The phases
and transitions are discussed in the text in detail.


# A characterization of $X$ for which spaces $C_{p}(X)$ are distinguished and
its applications

Jerzy Ka̧kol  Faculty of Mathematics and Informatics, A. Mickiewicz
University, 61-614 Poznań, Poland and Institute of Mathematics Czech Academy
of Sciences, Prague, Czech
Republic<EMAIL_ADDRESS>and  Arkady Leiderman  Department of Mathematics,
Ben-Gurion University of the Negev, Beer Sheva,
Israel<EMAIL_ADDRESS>
###### Abstract.

We prove that the locally convex space $C_{p}(X)$ of continuous real-valued
functions on a Tychonoff space $X$ equipped with the topology of pointwise
convergence is distinguished if and only if $X$ is a $\Delta$-space in the
sense of [21]. As an application of this characterization theorem we obtain
the following results:

1) If $X$ is a Čech-complete (in particular, compact) space such that
$C_{p}(X)$ is distinguished, then $X$ is scattered. 2) For every separable
compact space of the Isbell–Mrówka type $X$, the space $C_{p}(X)$ is
distinguished. 3) If $X$ is the compact space of ordinals $[0,\omega_{1}]$,
then $C_{p}(X)$ is not distinguished.

We observe that the existence of an uncountable separable metrizable space $X$
such that $C_{p}(X)$ is distinguished, is independent of ZFC. We explore also
the question to which extent the class of $\Delta$-spaces is invariant under
basic topological operations.

###### Key words and phrases:

Distinguished locally convex space, scattered compact space, $\Delta$-set,
Isbell–Mrówka space

###### 2010 Mathematics Subject Classification:

54C35, 54G12, 54H05, 46A03

The research for the first named author is supported by the GAČR project
20-22230L and RVO: 67985840. He thanks also the Center For Advanced Studies in
Mathematics of Ben-Gurion University of the Negev for financial support during
his visit in 2019.

##  1\. Introduction

Following J. Dieudonné and L. Schwartz [8] a locally convex space (lcs) $E$ is
called _distinguished_ if every bounded subset of the bidual of $E$ in the
weak∗-topology is contained in the closure of the weak∗-topology of some
bounded subset of $E$. Equivalently, a lcs $E$ is distinguished if and only if
the strong dual of $E$ (i.e. the topological dual of $E$ endowed with the
strong topology) is _barrelled_ , (see [19, 8.7.1]). A. Grothendieck [17]
proved that a metrizable lcs $E$ is distinguished if and only if its strong
dual is _bornological_. We refer the reader to survey articles [5] and [6]
which present several more modern results about distinguished metrizable and
Fréchet lcs.

Throughout the article, all topological spaces are assumed to be Tychonoff and
infinite. By $C_{p}(X)$ and $C_{k}(X)$ we mean the spaces of all real-valued
continuous functions on a Tychonoff space $X$ endowed with the topology of
pointwise convergence and the compact-open topology, respectively. By a
_bounded set_ in a topological vector space (in particular, $C_{p}(X)$) we
understand any set which is absorbed by every $0$-neighbourhood.

For spaces $C_{p}(X)$ we proved in [13] the following theorem (the equivalence
$(1)\Leftrightarrow(4)$ has been obtained in [11]).

######  Theorem 1.1.

For a Tychonoff space $X$, the following conditions are equivalent:

  1. (1)

$C_{p}(X)$ is distinguished.

  2. (2)

$C_{p}\left(X\right)$ is a large subspace of $\mathbb{R}^{X}$, i.e. for every
bounded set $A$ in $\mathbb{R}^{X}$ there exists a bounded set $B$ in
$C_{p}(X)$ such that $A\subset cl_{{\mathbb{R}}^{X}}(B)$.

  3. (3)

For every $f\in{\mathbb{R}}^{X}$ there is a bounded set $B\subset C_{p}(X)$
such that $f\in cl_{{\mathbb{R}}^{X}}(B)$.

  4. (4)

The strong dual of the space $C_{p}(X)$ carries the finest locally convex
topology.

Several examples of $C_{p}(X)$ with(out) distinguished property have been
provided in papers [11], [12] and [13]. The aim of this research is to
continue our initial work on distinguished spaces $C_{p}(X)$.

The following concept plays a key role in our paper. We show its applicability
for the studying of distinguished spaces $C_{p}(X)$.

######  Definition 1.2.

([21]) A topological space $X$ is said to be a $\Delta$-space if for every
decreasing sequence $\\{D_{n}:n\in\omega\\}$ of subsets of $X$ with empty
intersection, there is a decreasing sequence $\\{V_{n}:n\in\omega\\}$
consisting of open subsets of $X$, also with empty intersection, and such that
$D_{n}\subset V_{n}$ for every $n\in\omega$.

We should mention that R. W. Knight [21] called all topological spaces $X$
satisfying the above Definition 1.2 by $\Delta$-sets. The original definition
of a $\Delta$-set of the real line ${\mathbb{R}}$ is due to G. M. Reed and E.
K. van Douwen (see [27]). In this paper, for general topological spaces
satisfying Definition 1.2 we reserve the term _$\Delta$ -space_. The class of
all $\Delta$-spaces is denoted by $\Delta$.

In Section 2 we give an intrinsic description of all spaces $X\in\Delta$. One
of the main results of our paper, Theorem 2.1 says that $X$ is a
$\Delta$-space if and only if $C_{p}(X)$ is a distinguished space. This
characterization theorem has been applied systematically for obtaining a range
of results from our paper.

Our main result in Section 3 states that a Čech-complete (in particular,
compact) $X\in\Delta$ must be scattered. A very natural question arises what
are those scattered compact spaces $X\in\Delta$. In view of Theorem 2.1, it is
known that a Corson compact $X$ belongs to the class $\Delta$ if and only if
$X$ is a scattered Eberlein compact space [13]. With the help of Theorems 2.1
and 3.8 we show that the class $\Delta$ contains also all separable compact
spaces of the Isbell–Mrówka type. Nevertheless, as we demonstrate in Section
3, there are compact scattered spaces $X\notin\Delta$ (for example, the
compact space $[0,\omega_{1}]$).

Section 4 deals with the questions about metrizable spaces $X\in\Delta$. We
notice that every $\sigma$-scattered metrizable space $X$ belongs to the class
$\Delta$. For separable metrizable spaces $X$, our analysis reveals a tight
connection between distinguished $C_{p}(X)$ and well-known set-theoretic
problems about special subsets of the real line ${\mathbb{R}}$. We observe
that the existence of an uncountable separable metrizable space $X\in\Delta$
is independent of ZFC and it is equivalent to the existence of a separable
countably paracompact nonnormal Moore space. We refer readers to [24] for the
history of the normal Moore problem.

In Section 5 we study whether the class $\Delta$ is invariant under the basic
topological operations: subspaces, (quotient) continuous images,
finite/countable unions and finite products. We pose several new open
problems.

##  2\. Description Theorem

In this section we provide an intrinsic description of $X\in\Delta$. For the
reader’s convenience we recall some relevant terminology.

  1. (a)

A disjoint cover $\\{X_{\gamma}:\gamma\in\Gamma\\}$ of $X$ is called a
partition of $X$.

  2. (b)

A collection of sets $\\{U_{\gamma}:\gamma\in\Gamma\\}$ is called an expansion
of a collection of sets $\\{X_{\gamma}:\gamma\in\Gamma\\}$ in $X$ if
$X_{\gamma}\subseteq U_{\gamma}\subseteq X$ for every index $\gamma\in\Gamma$.

  3. (c)

A collection of sets $\\{U_{\gamma}:\gamma\in\Gamma\\}$ is called point-finite
if no point belongs to infinitely many $U_{\gamma}$ -s.

######  Theorem 2.1.

For a Tychonoff space $X$, the following conditions are equivalent:

  1. (1)

$C_{p}(X)$ is distinguished.

  2. (2)

Any countable partition of $X$ admits a point-finite open expansion in $X$.

  3. (3)

Any countable disjoint collection of subsets of $X$ admits a point-finite open
expansion in $X$.

  4. (4)

$X$ is a $\Delta$-space.

###### Proof.

Observe that every collection of pairwise disjoint subsets of $X$,
$\\{X_{\gamma}:\gamma\in\Gamma\\}$ can be extended to a partition by adding a
single set $X_{\ast}=X\setminus\bigcup\\{X_{\gamma}:\gamma\in\Gamma\\}$. If
the obtained partition admits a point-finite open expansion in $X$, then
removing one open set we get a point-finite open expansion of the original
disjoint collection. This shows evidently the equivalence (2)
$\Leftrightarrow$ (3).

Assume now that (3) holds. Let $\\{D_{n}:n\in\omega\\}$ be a decreasing
sequence subsets of $X$ with empty intersection. Define $X_{n}=D_{n}\setminus
D_{n+1}$ for each $n\in\omega$. By assumption, a disjoint collection
$\\{X_{n}:n\in\omega\\}$ admits a point-finite open expansion
$\\{U_{n}:n\in\omega\\}$ in $X$. Then $\\{V_{n}=\bigcup\\{U_{i}:i\geq
n\\}:n\in\omega\\}$ is an open decreasing expansion in $X$ with empty
intersection. This proves the implication (3) $\Rightarrow$ (4).

Next we show (4) $\Rightarrow$ (2). Let $\\{X_{n}:n\in\omega\\}$ be any
countable partition of $X$. Define $D_{0}=X$ and
$D_{n}=X\setminus\bigcup\\{X_{i}:i<n\\}$. Then $X_{n}\subset D_{n}$ for every
$n$, the sequence $\\{D_{n}:n\in\omega\\}$ is decreasing and its intersection
is empty. Assuming (4), we find an open decreasing expansion
$\\{U_{n}:n\in\omega\\}$ of $\\{D_{n}:n\in\omega\\}$ in $X$ such that
$\bigcap\\{U_{n}:n\in\omega\\}=\emptyset$. For every $x\in X$ there is $n$
such that $x\notin U_{m}$ for each $m>n$, it means that
$\\{U_{n}:n\in\omega\\}$ is a point-finite expansion of
$\\{X_{n}:n\in\omega\\}$ in $X$. This finishes the proof (3) $\Rightarrow$ (4)
$\Rightarrow$ (2) $\Leftrightarrow$ (3).

Now we prove the implication (1) $\Rightarrow$ (2). Let
$\\{X_{n}:n\in\omega\\}$ be any countable partition of $X$. Fix any function
$f\in{\mathbb{R}}^{X}$ which satisfies the following conditions: for each
$n\in\omega$ and every $x\in X_{n}$ the value of $f(x)$ is greater than $n$.
By assumption, there is a bounded subset $B$ of $C_{p}(X)$ such that $f\in
cl_{{\mathbb{R}}^{X}}(B)$. Hence, for every $n\in\omega$ and every point $x\in
X_{n}$, there exists $f_{x}\in B$ such that $f_{x}(x)>n$. But $f_{x}$ is a
continuous function, therefore there is an open neighbourhood $U_{x}\subset X$
of $x$ such that $f_{x}(y)>n$ for every $y\in U_{x}$. We define an open set
$U_{n}\subset X$ as follows: $U_{n}=\bigcup\\{U_{x}:x\in X_{n}\\}$. Evidently,
$X_{n}\subseteq U_{n}$ for each $n\in\omega$. If we assume that the open
expansion $\\{U_{n}:n\in\omega\\}$ is not point-finite, then there exists a
point $y\in X$ such that there are infinitely many numbers $n$ with $y\in
U_{x_{n}}$ for some $x_{n}\in X_{n}$. This means that $\sup\\{g(y):g\in
B\\}=\infty$, which contradicts the boundedness of $B$.

It remains to prove (2) $\Rightarrow$ (1). By Theorem 1.1, we need to show
that for every mapping $f\in{\mathbb{R}}^{X}$ there is a bounded set $B\subset
C_{p}(X)$ such that $f\in cl_{{\mathbb{R}}^{X}}(B)$. If there exists a
constant $r>0$ such that $\sup\\{|f(x)|:x\in X\\}<r$, then we take $B=\\{h\in
C(X):\sup\\{|h(x)|:x\in X\\}<r\\}$. It is easy to see that $B$ is as required.

Let $f\in{\mathbb{R}}^{X}$ be unbounded. Denote by $Y_{0}=\emptyset$ and
$Y_{n}=\\{x\in X:n-1\leq|f(x)|<n\\}$ for each non-zero $n\in\omega$. Define
$\varphi:X\rightarrow\omega$ by the rule: if $Y_{n}\neq\emptyset$ then
$\varphi(x)=n$ for every $x\in Y_{n}$. So, $|f|<\varphi$. Put
$X_{n}=\varphi^{-1}(n)$ for each $n\in\omega$. Note that some sets $X_{n}$
might happen to be empty, but the collection $\\{X_{n}:n\in\omega\\}$ is a
partition of $X$ with countably many nonempty $X_{n}$ -s. By our assumption,
there exists a point-finite open expansion $\\{U_{n}:n\in\omega\\}$ of the
partition $\\{X_{n}:n\in\omega\\}$. Define $F:X\rightarrow\omega$ by
$F(x)=\max\\{n:x\in U_{n}\\}$. Obviously, $f<F$. Finally, we define $B=\\{h\in
C_{p}(X):|h|\leq F\\}$. Then $f\in cl_{{\mathbb{R}}^{X}}(B)$, because for
every finite subset $K\subset X$ there is a function $h\in B$ such that
$f\restriction_{K}=h\restriction_{K}$. Indeed, given a finite subset $K\subset
X$, let $\\{V_{x}:x\in K\\}$ be the family of pairwise disjoint open sets such
that $x\in V_{x}\subset U_{\varphi(x)}$ for every $x\in K$. For each $x\in K$,
fix a continuous function $h_{x}:X\rightarrow[-\varphi(x),\varphi(x)]$ such
that $h_{x}(x)=f(x)$ and $h_{x}$ is equal to the constant value $0$ on the
closed set $X\setminus V_{x}$. One can verify that $h=\Sigma_{x\in K}h_{x}\in
B$ is as required. ∎

Below we present a straightforward application of Theorem 2.1.

######  Corollary 2.2.

([13]) Let $Z$ be any subspace of $X$. If $X$ belongs to the class $\Delta$,
then $Z$ also belongs to the class $\Delta$.

###### Proof.

If $\\{Z_{\gamma}:\gamma\in\Gamma\\}$ is any collection of pairwise disjoint
subsets of $Z$ and there exists a point-finite open expansion
$\\{U_{\gamma}:\gamma\in\Gamma\\}$ in $X$, then obviously $\\{U_{\gamma}\cap
Z:\gamma\in\Gamma\\}$ is a point-finite expansion consisting of the sets
relatively open in $Z$. It remains to apply Theorem 2.1. ∎

The last result can be reversed, assuming that $X\setminus Z$ is finite.

######  Proposition 2.3.

Let $Z$ be a subspace of $X$ such that $Y=X\setminus Z$ is finite. If $Z$
belongs to the class $\Delta$, then $X$ belongs to $\Delta$ as well.

###### Proof.

Let $\\{X_{n}:n\in\omega\\}$ be any countable collection of pairwise disjoint
subsets of $X$. Denote by $F$ the set of those $n\in\omega$ such that
$X_{n}\bigcap Y\neq\emptyset$. There might be only finitely many $X_{n}$ -s
which intersect the finite set $Y$, hence $F\subset\omega$ is finite. If $n\in
F$, then we simply declare that $U_{n}$ is equal to $X$. Consider the
subcollection $\\{X_{n}:n\in\omega\setminus F\\}$. It is a countable
collection of pairwise disjoint subsets of $Z$. Since $Z\in\Delta$, by Theorem
2.1, there is a point-finite open expansion $\\{U_{n}:n\in\omega\setminus
F\\}$ in $Z$. Observe that $Z$ is open in $X$, therefore all those $U_{n}$ -s
remain open in $X$. Bringing all $U_{n}$ -s of both sorts together we obtain a
point-finite open expansion $\\{U_{n}:n\in\omega\\}$ in $X$. Finally,
$X\in\Delta$, by Theorem 2.1. ∎

######  Remark 2.4.

The following applicable concept has been re-introduced in [13]. A family
$\left\\{\mathcal{N}_{x}:x\in X\right\\}$ of subsets of a Tychonoff space $X$
is called a _scant cover_ for $X$ if each $\mathcal{N}_{x}$ is an open
neighbourhood of $x$ and for each $u\in X$ the set $X_{u}=\left\\{x\in
X:u\in\mathcal{N}_{x}\right\\}$ is finite. 111 The referee kindly informed the
authors that this notion also is known in the literature under the name _the
point-finite neighbourhood assignment_.

Our Theorem 2.1 generalizes one of the results obtained in [13] stating that
if $X$ admits a scant cover $\left\\{\mathcal{N}_{x}:x\in X\right\\}$ then
$C_{p}\left(X\right)$ is distinguished. Indeed, let
$\\{X_{\gamma}:\gamma\in\Gamma\\}$ be any collection of pairwise disjoint
subsets of $X$. Define $U_{\gamma}=\bigcup\\{\mathcal{N}_{x}:x\in
X_{\gamma}\\}$. It is easily seen that $\\{U_{\gamma}:\gamma\in\Gamma\\}$ is a
point-finite open expansion in $X$, by definition of a scant cover. Applying
Theorem 2.1, we conclude that $C_{p}\left(X\right)$ is distinguished.

##  3\. Applications to compact spaces $X\in\Delta$

First we recall a few definitions and facts (probably well-known) which will
be used in the sequel. A space $X$ is said to be scattered if every nonempty
subset $A$ of $X$ has an isolated point in $A$. Denote by $A^{(1)}$ the set of
all non-isolated (in $A$) points of $A\subset X$. For ordinal numbers
$\alpha$, the $\alpha$-th derivative of a topological space $X$ is defined by
transfinite induction as follows.

$X^{(0)}=X$; $X^{(\alpha+1)}=(X^{(\alpha)})^{(1)}$;
$X^{(\gamma)}=\bigcap_{\alpha<\gamma}X^{(\alpha)}$ for limit ordinals
$\gamma$.

For a scattered space $X$, the smallest ordinal $\alpha$ such that
$X^{(\alpha)}=\emptyset$ is called the _scattered height_ of $X$ and is
denoted by $ht(X)$. For instance, $X$ is discrete if and only if $ht(X)=1$.

The following classical theorem is due to A. Pełczyński and Z. Semadeni.

######  Theorem 3.1.

([28, Theorem 8.5.4]) A compact space $X$ is scattered if and only if there is
no continuous mapping of $X$ onto the segment $[0,1]$.

A continuous surjection $\pi:X\rightarrow Y$ is called irreducible (see [28,
Definition 7.1.11]) if for every closed subset $F$ of $X$ the condition
$\pi(F)=Y$ implies $F=X$.

######  Proposition 3.2.

([28, Proposition 7.1.13]) Let $X$ be a compact space and let
$\pi:X\rightarrow Y$ be a continuous surjection. Then there exists a closed
subset $F$ of $X$ such that $\pi(F)=Y$ and the restriction
${\pi\res_{F}:F\rightarrow Y}$ is irreducible.

######  Proposition 3.3.

([28, Proposition 25.2.1]) Let $X$ be a compact space and let
$\pi:X\rightarrow Y$ be a continuous surjection. Then $\pi$ is irreducible if
and only if whenever $E\subset X$ and $\pi(E)$ is dense in $Y$, then $E$ is
dense in $X$.

Recall that a Tychonoff space $X$ is _Čech-complete_ if $X$ is a
$G_{\delta}$-set in some (equivalently, any) compactification of $X$, (see [9,
3.9.1]). It is well known that every locally compact space and every
completely metrizable space is Čech-complete. Next statement resolves an open
question posed in [13].

######  Theorem 3.4.

Every Čech-complete (in particular, compact) $\Delta$-space is scattered.

###### Proof.

_Step 1: $X$ is compact_. On the contrary, assume that $X$ is not scattered.
First, by Theorem 3.1, there is a continuous mapping $\pi$ from $X$ onto the
segment $[0,1]$. Second, by Proposition 3.2, there exists a closed subset $F$
of $X$ such that $\pi(F)=[0,1]$ and the restriction
${\pi\res_{F}:F\rightarrow[0,1]}$ is irreducible. Since $X\in\Delta$ the
compact space $F$ also belongs to $\Delta$, by Corollary 2.2. For simplicity,
without loss of generality we may assume that $F$ is $X$ itself and
$\pi:X\rightarrow[0,1]$ is irreducible.

Let $\\{X_{n}:n\in\omega\\}$ be a partition of $[0,1]$ into dense sets. Put
$Y_{n}=\bigcup_{k\geq n}X_{k}$, and $Z_{n}=\pi^{-1}(Y_{n})$ for all
$n\in\omega$. Then all sets $Z_{n}$ are dense in $X$ by Proposition 3.3 and
the intersection $\bigcap_{n\in\omega}Z_{n}$ is empty. Every compact space $X$
is a Baire space, i.e. the Baire category theorem holds in $X$, hence if
$\\{U_{n}:n\in\omega\\}$ is any open expansion of $\\{Z_{n}:n\in\omega\\}$,
then the intersection $\bigcap_{n\in\omega}U_{n}$ is dense in $X$. In view of
our Theorem 2.1 this conclusion contradicts the assumption $X\in\Delta$, and
the proof follows.

_Step 2: $X$ is any Čech-complete space_. By the first step we deduce that
every compact subset of $X$ is scattered. But any Čech-complete space $X$ is
scattered if and only if every compact subset of $X$ is scattered. A detailed
proof of this probably folklore statement can be found in [29]. ∎

######  Proposition 3.5.

If $X$ is a first-countable compact space, then $X\in\Delta$ if and only if
$X$ is countable.

###### Proof.

If $X\in\Delta$, then $X$ is scattered, by Theorem 3.4. By the classical
theorem of S. Mazurkiewicz and W. Sierpiński [28, Theorem 8.6.10], a first-
countable compact space is scattered if and only if it is countable. This
proves (i) $\Rightarrow$ (ii). The converse is known [13] and follows from the
fact that any countable space $X=\\{x_{n}:n\in\omega\\}$ admits a scant cover.
Indeed, define $X_{n}=\\{x_{i}:i\geq n\\}$. Then the family
$\\{X_{n}:n\in\omega\\}$ is a scant cover of $X$. Now it suffices to mention
Remark 2.4. ∎

######  Remark 3.6.

Theorem 3.4 extends also a well-known result of B. Knaster and K. Urbanik
stating that every countable Čech-complete space is scattered [20]. It is easy
to see that a countable Baire space contains a dense subset of isolated
points, but in general does not have to be scattered. We don’t know whether
every Baire $\Delta$-space must have isolated points.

Recall that an Eberlein compact is a compact space homeomorphic to a subset of
a Banach space with the weak topology. A compact space is said to be a Corson
compact space if it can be embedded in a $\Sigma$-product of the real lines.
Every Eberlein compact is Corson, but not vice versa. However, every scattered
Corson compact space is a scattered Eberlein compact space [1].

######  Theorem 3.7.

([13]) A Corson compact space $X$ belongs to the class $\Delta$ if and only if
$X$ is a scattered Eberlein compact space.

Bearing in mind Theorem 3.4, to show Theorem 3.7 it suffices to use the fact
that every scattered Eberlein compact space admits a scant cover (the latter
follows from the proof of [4, Lemma 1.1]) and then apply Remark 2.4.

Being motivated by the previous results one can ask if there exist scattered
compact spaces $X\in\Delta$ which are not Eberlein compact. The next question
is also crucial: Does there exist a compact scattered space $X\notin\Delta$?
Below we answer both questions positively.

We need the following somewhat technical

######  Theorem 3.8.

Let $Z=C_{0}\cup C_{1}$ be a Tychonoff space such that

  1. (1)

$C_{0}\cap C_{1}=\emptyset$.

  2. (2)

$C_{0}$ is an open $F_{\sigma}$ subset of $Z$.

  3. (3)

both $C_{0}$ and $C_{1}$ belong to the class $\Delta$.

Then $Z$ also belongs to the class $\Delta$.

###### Proof.

By assumption, $C_{0}=\bigcup\\{F_{n}:n\in\omega\\}$, where each $F_{n}$ is
closed in $Z$. Let $\\{X_{n}:n\in\omega\\}$ be any countable collection of
pairwise disjoint subsets of $Z$. Our target is to define open sets
$U_{n}\supseteq X_{n}$, $n\in\omega$ in such a way that the collection
$\\{U_{n}:n\in\omega\\}$ is point-finite. We decompose the sets
$X_{n}=X_{n}^{0}\cup X_{n}^{1}$, where $X_{n}^{0}=X_{n}\cap C_{0}$ and
$X_{n}^{1}=X_{n}\cap C_{1}$. By Theorem 2.1, the collection
$\\{X_{n}^{0}:n\in\omega\\}$ expands to a point-finite open collection
$\\{U_{n}^{0}:n\in\omega\\}$ in $C_{0}$. The set $C_{0}$ is open in $Z$,
therefore $U_{n}^{0}$ are open in $Z$ as well.

Now we consider the disjoint collection $\\{X_{n}^{1}:n\in\omega\\}$ in
$C_{1}$. By assumption, $C_{1}\in\Delta$, therefore applying Theorem 2.1 once
more, we find a point-finite expansion $\\{V_{n}^{1}:n\in\omega\\}$ in $C_{1}$
consisting of sets which are open in $C_{1}$. Every set $V_{n}^{1}$ is a trace
of some set $W_{n}^{1}$, which is open in $Z$, i.e. $V_{n}^{1}=W_{n}^{1}\cap
C_{1}$, and every $W_{n}^{1}$ is open in $Z$. We refine the sets $W_{n}^{1}$
by the formula $U_{n}^{1}=W_{n}^{1}\setminus\bigcup\\{F_{i}:i\leq n\\}$. Since
all sets $F_{i}$ are closed in $Z$, the sets $U_{n}^{1}$ remain open in $Z$.
Since all sets $F_{i}$ are disjoint with $C_{1}$, the collection
$\\{U_{n}^{1}:n\in\omega\\}$ remains to be an expansion of
$\\{X_{n}^{1}:n\in\omega\\}$. Furthermore, the collection
$\\{U_{n}^{1}:n\in\omega\\}$ is point-finite, because
$\\{V_{n}^{1}:n\in\omega\\}$ is point-finite, and every point $z\in C_{0}$
belongs to some $F_{n}$, hence $z\notin U_{m}^{1}$ for every $m\geq n$.
Finally, we define $U_{n}=U_{n}^{0}\cup U_{n}^{1}$. The collection
$\\{U_{n}:n\in\omega\\}$ is a point-finite open expansion of
$\\{X_{n}:n\in\omega\\}$, and the proof is complete. ∎

This yields the following

######  Corollary 3.9.

Let $Z$ be any separable scattered Tychonoff space such that its scattered
height $ht(Z)$ is equal to 2. Then $Z\in\Delta$.

###### Proof.

The structure of $Z$ is the following. $Z=C_{0}\cup C_{1}$, where $C_{0}$ is a
countable dense in $Z$ set consisting of isolated in $Z$ points and $C_{1}$
consists of all accumulation points. Moreover, the space $C_{1}$ with the
topology induced from $Z$ is discrete. All conditions of Theorem 3.8 are
satisfied, and the result follows. ∎

Our first example will be the one-point compactification of an Isbell–Mrówka
space $\Psi({\mathcal{A}})$. We recall the construction and basic properties
of $\Psi({\mathcal{A}})$. Let ${\mathcal{A}}$ be an almost disjoint family of
subsets of the set of natural numbers ${\mathbb{N}}$ and let
$\Psi({\mathcal{A}})$ be the set ${\mathbb{N}}\cup{\mathcal{A}}$ equipped with
the topology defined as follows. For each $n\in{\mathbb{N}}$, the singleton
$\\{n\\}$ is open, and for each $A\in{\mathcal{A}}$, a base of neighbourhoods
of $A$ is the collection of all sets of the form $\\{A\\}\cup B$, where
$B\subset A$ and $|A\setminus B|<\omega$. The space $\Psi({\mathcal{A}})$ is
then a first-countable separable locally compact Tychonoff space. If
${\mathcal{A}}$ is a maximal almost disjoint (MAD) family, then the
corresponding Isbell–Mrówka space $\Psi({\mathcal{A}})$ would be in addition
pseudocompact. (Readers are advised to consult [18, Chapter 8] which surveys
various topological properties of these spaces).

######  Theorem 3.10.

There exists a separable scattered compact space $X$ with the following
properties:

  1. (a)

The scattered height of $X$ is equal to 3.

  2. (b)

$X\in\Delta$.

  3. (c)

$X$ is not an Eberlein compact space.

###### Proof.

Let ${\mathcal{A}}$ be any uncountable almost disjoint (in particular, MAD)
family of subsets of ${\mathbb{N}}$ and let $Z$ be the corresponding first-
countable separable locally compact Isbell–Mrówka space $\Psi({\mathcal{A}})$.
It is easy to see that $Z=\Psi({\mathcal{A}})$ satisfies the assumptions of
Corollary 3.9. Hence, $Z\in\Delta$. Now, denote by $X$ the one-point
compactification of the separable locally compact space $Z$. Then the
scattered height of $X$ is equal to 3. Note that $X\in\Delta$ by Proposition
2.3. Moreover, $X$ is not an Eberlein compact space, since every separable
Eberlein compact space is metrizable, while $\Psi({\mathcal{A}})$ is
metrizable if and only if ${\mathcal{A}}$ is countable. ∎

Now we show that there exist scattered compact spaces which are not in the
class $\Delta$. We will use the classical Pressing Down Lemma. Let
$[0,\omega_{1})$ be the set of all countable ordinals equipped with the order
topology. For simplicity, we identify $[0,\omega_{1})$ with $\omega_{1}$. A
subset $S$ of $\omega_{1}$ is called a stationary subset if $S$ has nonempty
intersection with every closed and unbounded set in $\omega_{1}$. A mapping
$\varphi:S\rightarrow\omega_{1}$ is called regressive if
$\varphi(\alpha)<\alpha$ for each $\alpha\in S$. The proof of the following
fundamental statement can be found for instance in [22].

######  Theorem 3.11.

Pressing Down Lemma. Let $\varphi:S\rightarrow\omega_{1}$ be a regressive
mapping, where $S$ is a stationary subset of $\omega_{1}$. Then for some
$\gamma<\omega_{1}$, $\varphi^{-1}\\{{\gamma}\\}$ is a stationary subset of
$\omega_{1}$.

It is known that there are plenty of stationary subsets of $\omega_{1}$. In
particular, every stationary set can be partitioned into countably many
pairwise disjoint stationary sets [22]. Note that $\omega_{1}$ is a scattered
locally compact and first-countable space. Next statement resolves an open
question posed in [13].

######  Theorem 3.12.

The compact scattered space $[0,\omega_{1}]$ is not in the class $\Delta$.

###### Proof.

It suffices to show that $\omega_{1}$ does not belong to the class $\Delta$.
Assume, on the contrary, that $\omega_{1}\in\Delta$. Denote by $L$ the set of
all countable limit ordinals. Evidently, $L$ is a closed unbounded set in
$\omega_{1}$. Take any representation of $L$ as the union of countably many
pairwise disjoint stationary sets $\\{S_{n}:n\in\omega\\}$. By Theorem 2.1,
there exists a point-finite open expansion $\\{U_{n}:n\in\omega\\}$ in
$\omega_{1}$.

Fr every $\alpha\in U_{n}$ there is an ordinal $\beta(\alpha)<\alpha$ such
that $[\beta(\alpha),\alpha]\subset U_{n}$. In fact, for every $n\in\omega$ we
can define a regressive mapping $\varphi_{n}:S_{n}\rightarrow\omega_{1}$ by
the formula: $\varphi_{n}(\alpha)=\beta(\alpha)$. Since $S_{n}$ is a
stationary set for every $n$, we can apply to $\varphi_{n}$ the Pressing Down
Lemma. Hence, for each $n$ there are a countable ordinal $\gamma_{n}$ and an
uncountable subset $T_{n}\subset S_{n}$ with the following property:
$[\gamma_{n},\alpha]\subset U_{n}$ for every $\alpha\in T_{n}$. Denote
$\gamma=\sup\\{\gamma_{n}:n\in\omega\\}\in\omega_{1}$. Because all $T_{n}$ are
unbounded, for all natural $n$ we have an ordinal $\alpha_{n}\in T_{n}$ such
that $\gamma<\alpha_{n}$ and $[\gamma_{n},\alpha_{n}]\subset U_{n}$. This
implies that $\gamma\in U_{n}$ for every $n\in\omega$. However, a collection
$\\{U_{n}:n\in\omega\\}$ is point-finite. The obtained contradiction finishes
the proof. ∎

The function space $C_{k}(X)$ is called _Asplund_ if every separable vector
subspace of $C_{k}(X)$ isomorphic to a Banach space, has the separable dual.

######  Proposition 3.13.

If a Tychonoff space $X$ belongs to the class $\Delta$, then the space
$C_{k}(X)$ is Asplund. The converse conclusion fails in general.

###### Proof.

Let $\mathcal{K}(X)$ be the family of all compact subset of $X$. By the
assumption and Corollary 2.2, each $K\in\mathcal{K}(X)$ belongs to the class
$\Delta$. Clearly, $C_{k}(X)$ is isomorphic to a (closed) subspace of the
product $\Pi=\prod_{K\in\mathcal{K}(X)}C_{k}(K)$ of Banach spaces $C_{k}(K)$.
Assume that $E$ is a separable vector subspace of $C_{k}(X)$ isomorphic to a
Banach space. Observe that $E$ is isomorphic to a subspace of the finite
product $\prod_{j\in F}C_{k}(K_{j})$ for $K_{j}\in\mathcal{K}(X)$ and $j\in
F$. Indeed, let $B$ be the unit (bounded) ball of the normed space $E$. Then
there exists a finite set $F$ such that $\bigcap_{j\in
F}\pi^{-1}_{j}(U_{j})\cap\Pi\subset B$, where $U_{j}$ are balls in spaces
$C_{k}(K_{j})$, $j\in F$, and $\pi_{j}$ are natural projections from $E$ onto
$C_{k}(K_{j})$. Let $\pi_{F}$ be the (continuous) projection from $\Pi$ onto
$\prod_{\j\in F}C_{k}(K_{j})$. Then $\pi_{F}\restriction_{E}$ is an injective
continuous and open map from $E$ onto
$(\pi_{F}\restriction_{E})(E)\subset\prod_{\j\in F}C_{k}(K_{j})$. The
injectivity of $\pi_{F}\restriction_{E}$ follows from the fact that $B$ is a
bounded neighbourhood of zero in $E$. It is easy to see that the image
$(\pi_{F}\restriction_{E})(B)$ is an open neighbourhood of zero in
$\prod_{j\in F}C_{k}(K_{j})$. On the other hand, $\prod_{j\in F}C_{k}(K_{j})$
is isomorphic to the space $C_{k}(\bigoplus_{j\in F}K_{j})$ and the compact
space $\bigoplus_{j\in F}K_{j}$ is scattered. By the classical [10, Theorem
12.29] $E$ must have the separable dual $E^{*}$. Hence, $C_{k}(X)$ is Asplund.
The converse fails, as Theorem 3.12 shows for $X=[0,\omega_{1}]$. ∎

Since every infinite compact scattered space $X$ contains a nontrivial
converging sequence, for such $X$ the Banach space $C(X)$ is not a
Grothendieck space, (see [7]).

######  Corollary 3.14.

If $X$ is an infinite compact and $X\in\Delta$, then the Banach space $C(X)$
is not a Grothendieck space. The converse fails, as $X=[0,\omega_{1}]$
applies.

For non-scattered spaces $X$ Theorem 3.4 implies immediately the following

######  Corollary 3.15.

If $X$ is a non-scattered space, the Stone-Čech compactification $\beta X$ is
not in the class $\Delta$.

######  Proposition 3.16.

Let $X=\beta Z\setminus Z$, where $Z$ is any infinite discrete space. Then $X$
is not in the class $\Delta$.

###### Proof.

$\beta Z\setminus Z$ does not have isolated points for any infinite discrete
space $Z$. ∎

It is known that $X=[0,\omega_{1}]$ is the Stone-Čech compactification of
$[0,\omega_{1})$. We showed that $X\notin\Delta$. Also, $\beta Z\notin\Delta$
for any infinite discrete space $Z$. Every scattered Eberlein compact space
belongs to the class $\Delta$ by Theorem 3.7, however no Eberlein compact $X$
can be the Stone-Čech compactification $\beta Z$ for any proper subset $Z$ of
$X$ by the Preiss–Simon theorem (see [2, Theorem IV.5.8]). All these facts
provides a motivation for the following result.

######  Example 3.17.

There exists an Isbell–Mrówka space $Z$ which is almost compact in the sense
that the one-point compactification of $Z$ coincides with $\beta Z$ (see [18,
Theorem 8.6.1]). Define $X=\beta Z$. Then $X\in\Delta$, by Theorem 3.10.

##  4\. Metrizable spaces $X\in\Delta$

In this section we try to describe constructively the structure of nontrivial
metrizable spaces $X\in\Delta$. Note first that every scattered metrizable $X$
is in the class $\Delta$ since every such space $X$ homeomorphically embeds
into a scattered Eberlein compact [3], and then Theorem 3.7 and Corollary 2.2
apply. We extend this result as follows.

A topological space $X$ is said to be _$\sigma$ -scattered_ if $X$ can be
represented as a countable union of scattered subspaces and $X$ is called
_strongly $\sigma$-discrete_ if it is a union of countably many of its closed
discrete subspaces. Strongly $\sigma$-discreteness of $X$ implies that $X$ is
$\sigma$-scattered, for any topological space. For metrizable $X$, by the
classical result of A. H. Stone [30], these two properties are equivalent.

######  Proposition 4.1.

Any $\sigma$-scattered metrizable space belongs to the class $\Delta$.

###### Proof.

In view of aforementioned equivalence, every subset of $X$ is $F_{\sigma}$. If
every subset of $X$ is $F_{\sigma}$, then $X\in\Delta$. This fact apparently
is well-known (see also a comment after Claim 4.2). For the sake of
completeness we include a direct argument. We show that $X$ satisfies the
condition (2) of Theorem 2.1. Let $\\{X_{n}:n\in\omega\\}$ be any countable
disjoint partition of $X$. Denote $X_{n}=\bigcup\\{F_{n,m}:m\in\omega\\}$,
where each $F_{n,m}$ is closed in $X$. Define open sets $U_{n}$ as follows:
$U_{0}=X$ and $U_{n}=X\setminus\bigcup\\{F_{k,m}:k<n,m<n\\}$ for $n\geq 1$.
Then $\\{U_{n}:n\in\omega\\}$ is a point-finite open expansion of
$\\{X_{n}:n\in\omega\\}$ in $X$. ∎

A metrizable space $A$ is called an _absolutely analytic_ if $A$ is
homeomorphic to a Souslin subspace of a complete metric space $X$ (of an
arbitrary weight), i.e. $A$ is expressible as
$A=\bigcup_{\sigma\in{\mathbb{N}}^{\mathbb{N}}}\bigcap_{n\in{\mathbb{N}}}A_{\sigma|n}$,
where each $A_{\sigma|n}$ is a closed subset of $X$. It is known that every
absolutely analytic metrizable space $X$ (in particular, every Borel subspace
of a complete metric space) either contains a homeomorphic copy of the Cantor
set or it is strongly $\sigma$-discrete. Therefore, for absolutely analytic
metrizable space $X$ the converse is true: $X\in\Delta$ implies that $X$ is
strongly $\sigma$-discrete [13].

However, the last structural result can not be proved in general for all
(separable) metrizable spaces without extra set-theoretic assumptions. Let us
recall several definitions of special subsets of the real line ${\mathbb{R}}$
(see [23], [27]).

  1. (a)

A $Q$-set $X$ is a subset of ${\mathbb{R}}$ such that each subset of $X$ is
$F_{\sigma}$, or, equivalently, each subset of $X$ is $G_{\delta}$ in $X$.

  2. (b)

A $\lambda$-set $X$ is a subset of ${\mathbb{R}}$ such that each countable
$A\subset X$ is $G_{\delta}$ in $X$.

  3. (c)

A $\Delta$-set $X$ is a subset of ${\mathbb{R}}$ such that for every
decreasing sequence $\\{D_{n}:n\in\omega\\}$ subsets of $X$ with empty
intersection there is a decreasing expansion $\\{V_{n}:n\in\omega\\}$
consisting of open subsets of $X$ with empty intersection.

######  Claim 4.2.

The existence of an uncountable separable metrizable $\Delta$-space is
equivalent to the existence of an uncountable $\Delta$-set.

###### Proof.

Note that every separable metrizable space homeomorphically embeds into a
Polish space ${\mathbb{R}}^{\omega}$ and the latter space is a one-to-one
continuous image of the set of irrationals ${\mathbb{P}}$. Therefore, if $M$
is an uncountable separable metrizable space, then there exist an uncountable
set $X\subset{\mathbb{R}}$ and a one-to-one continuous mapping from $X$ onto
$M$. It is easy to see that $X$ is a $\Delta$-set provided $M$ is a
$\Delta$-space. ∎

Note that in the original definition of a $\Delta$-set, G. M. Reed used
$G_{\delta}$-sets instead of open sets and E. van Douwen observed that these
two versions are equivalent [27]. From the original definition it is obvious
that each $Q$-set must be a $\Delta$-set. The fact that every $\Delta$-set is
a $\lambda$-set is known as well. K. Kuratowski showed that in ZFC there exist
uncountable $\lambda$-sets. The existence of an uncountable $Q$-set is one of
the fundamental set-theoretical problems considered by many authors. F.
Hausdorff showed that the cardinality of an uncountable $Q$-set $X$ has to be
strictly smaller than the continuum $\mathfrak{c}=2^{\aleph_{0}}$, so in
models of ZFC plus the Continuum Hypothesis (CH) there are no uncountable
$Q$-sets. Let us outline several known most relevant facts.

(1) Martin’s Axiom plus the negation of the Continuum Hypothesis (MA
$+\lnot$CH) implies that every subset $X\subset{\mathbb{R}}$ of cardinality
less than $\mathfrak{c}$ is a $Q$-set (see [15]).

(2) It is consistent that there is a $Q$-set $X$ such that its square $X^{2}$
is not a $Q$-set [14].

(3) The existence of an uncountable $Q$-set is equivalent to the existence of
an uncountable strong $Q$-set, i.e. a $Q$-set all finite powers of which are
$Q$-sets [25].

(4) No $\Delta$-set $X$ can have cardinality $\mathfrak{c}$ [26]. Hence, under
MA, every subset of ${\mathbb{R}}$ that is a $\Delta$-set is also a $Q$-set.
Recently we proved the following claim: If $X$ has a countable network and
$|X|=\mathfrak{c}$, then $C_{p}(X)$ is not distinguished [13]. In view of our
Theorem 2.1 this fact means that no $\Delta$-space $X$ with a countable
network can have cardinality $\mathfrak{c}$. 222 The referee kindly informed
the authors that the last result can be derived easily from the actual
argument of [26].

(5) It is consistent that there exists a $\Delta$-set $X$ that is not a
$Q$-set [21]. Of course, there are plenty of nonmetrizable $\Delta$-spaces
with non-$G_{\delta}$ subsets, in ZFC.

(6) An uncountable $\Delta$-set exists if and only if there exists a separable
countably paracompact nonnormal Moore space (see [16] and [26]).

Summarizing, the following conclusion is an immediate consequence of our
Theorem 2.1 and the known facts about $\Delta$-sets listed above.

######  Corollary 4.3.

  1. (1)

The existence of an uncountable separable metrizable space such that
$C_{p}(X)$ is distinguished, is independent of ZFC.

  2. (2)

There exists an uncountable separable metrizable space $X$ such that
$C_{p}(X)$ is distinguished, if and only if there exists a separable countably
paracompact nonnormal Moore space.

##  5\. Basic operations in $\Delta$ and open problems

In this section we consider the question whether the class $\Delta$ is
invariant under the following basic topological operations: subspaces,
continuous images, quotient continuous images, finite/countable unions, finite
products.

_1\. Subspaces_. Trivial because of Corollary 2.2.

_2\. (Quotient) continuous images_. Evidently, every topological space is a
continuous image of a discrete one. The following assertion is a consequence
of a known fact about MAD families (see [18, Chapter 8]).

######  Proposition 5.1.

There exists a first-countable separable pseudocompact locally compact
Isbell–Mrówka space $\Psi({\mathcal{A}})$ which admits a continuous surjection
onto the segment $[0,1]$.

Thus, the class $\Delta$ is not invariant under continuous images even for
separable locally compact spaces. However, one can show that every uncountable
quotient continuous image of any Isbell–Mrówka space $\Psi({\mathcal{A}})$
satisfies the conditions of Corollary 3.9, therefore it is a $\Delta$-space.
Note also that a class of scattered Eberlein compact spaces preserves
continuous images. We were unable to resolve the following major open problem.

######  Problem 5.2.

Let $X$ be any compact $\Delta$-space and $Y$ be a continuous image of $X$. Is
$Y$ a $\Delta$-space?

Even a more general question is open.

######  Problem 5.3.

Let $X$ be any $\Delta$-space and $Y$ be a quotient continuous image of $X$.
Is $Y$ a $\Delta$-space?

Towards a solution of these problems we obtained several partial positive
results.

######  Proposition 5.4.

Let $X$ be any $\Delta$-space and $\varphi:X\to Y$ be a quotient continuous
surjection with only finitely many nontrivial fibers. Then $Y$ is also a
$\Delta$-space.

###### Proof.

By assumption, there exists a closed subset $K\subset X$ such that
$\varphi(K)$ is finite and $\varphi\restriction_{X\setminus K}:X\setminus K\to
Y\setminus\varphi(K)$ is a one-to-one mapping. Both sets $X\setminus K$ and
$Y\setminus\varphi(K)$ are open in $X$ and $Y$, respectively. Since $\varphi$
is a quotient continuous mapping, it is easy to see that
$\varphi\restriction_{X\setminus K}$ is a homeomorphism. $X\setminus K$ is a
$\Delta$-space, hence $Y\setminus\varphi(K)$ is also a $\Delta$-space.
Finally, $Y$ is a $\Delta$-space, by Proposition 2.3. ∎

######  Proposition 5.5.

Let $X$ be any $\Delta$-space and $\varphi:X\to Y$ be a closed continuous
surjection with finite fibers. Then $Y$ is also a $\Delta$-space.

###### Proof.

Let $\\{Y_{n}:n\in\omega\\}$ be a partition of $Y$. By assumption, the
partition $\\{\varphi^{-1}(Y_{n}):n\in\omega\\}$ admits a point-finite open
expansion $\\{U_{n}:n\in\omega\\}$ in $X$. Clearly, $\varphi(X\setminus
U_{n})$ are closed sets in $Y$. Define $V_{n}=Y\setminus\varphi(X\setminus
U_{n})$ for each $n\in\omega$. We have that $\\{V_{n}:n\in\omega\\}$ is an
open expansion of $\\{Y_{n}:n\in\omega\\}$ in $Y$. It remains to verify that
the family $\\{V_{n}:n\in\omega\\}$ is point-finite. Indeed, let $y\in Y$ be
any point. Each point in the fiber $\varphi^{-1}(y)$ belongs to a finite
number of sets $U_{n}$. Since the fiber $\varphi^{-1}(y)$ is finite, $y$ is
contained only in a finite number of sets $V_{n}$ which finishes the proof. ∎

_3\. Finite/countable unions_.

######  Proposition 5.6.

Assume that $X$ is a finite union of closed subsets $X_{i}$, where each
$X_{i}$ belongs to the class $\Delta$. Then $X$ also belongs to $\Delta$. In
particular, a finite union of compact $\Delta$-spaces is also a
$\Delta$-space.

###### Proof.

Denote by $Z$ the discrete finite union of $\Delta$-spaces $X_{i}$. Obviously,
$Z$ is a $\Delta$-space which admits a natural closed continuous mapping onto
$X$. Since all fibers of this mapping are finite, the result follows from
Proposition 5.5. ∎

We recall a definition of the Michael line. The Michael line $X$ is the
refinement of the real line ${\mathbb{R}}$ obtained by isolating all
irrational points. So, $X$ can be represented as a countable disjoint union of
singletons (rationals) and an open discrete set. Nevertheless, the Michael
line $X$ is not in $\Delta$ [13]. This example and Proposition 5.6 justify the
following

######  Problem 5.7.

Let $X$ be a countable union of compact subspaces $X_{i}$ such that each
$X_{i}$ belongs to the class $\Delta$. Does $X$ belong to the class $\Delta$?

_4\. Finite products_. We already mentioned earlier that the existence of a
$Q$-set $X\subset{\mathbb{R}}$ such that its square $X^{2}$ is not a $Q$-set,
is consistent with ZFC.

######  Problem 5.8.

Is the existence of a $\Delta$-set $X\subset{\mathbb{R}}$ such that its square
$X^{2}$ is not a $\Delta$-set, consistent with ZFC?

It is known that the finite product of scattered Eberlein compact spaces is a
scattered Eberlein compact.

######  Problem 5.9.

Let $X$ be the product of two compact spaces $X_{1}$ and $X_{2}$ such that
each $X_{i}$ belongs to the class $\Delta$. Does $X$ belong to the class
$\Delta$?

Our last problem is inspired by Theorem 3.10.

######  Problem 5.10.

Let $X$ be any scattered compact space with a finite scattered height. Does
$X$ belong to the class $\Delta$?

Acknowledgements. The authors thank Michael Hrušák for a useful information
about Isbell–Mrówka spaces.

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