# Unidirectional mode selection in bistable quantum cascade ring lasers

Sara Kacmoli1,*, Deborah L. Sivco2, Claire F. Gmachl1

( 1Department of Electrical and Computer Engineering, Princeton University,
Princeton, NJ 08544 USA
2Current address: Trumpf Photonics, Inc., 2601 US Highway 130, Cranbury, NJ
08512 USA
<EMAIL_ADDRESS>
)

###### Abstract

Ideal ring resonators are characterized by travelling-wave counterpropagating
modes, but in practice travelling waves can only be realized under
unidirectional operation, which has proved elusive. Here, we have designed and
fabricated a monolithic quantum cascade ring laser coupled to an active
waveguide that allows for robust, deterministic and controllable
unidirectional operation. Spontaneous emission injection through the active
waveguide enables dynamical switching between the clockwise and
counterclockwise states of the ring laser with as little as 1.6$\%$ modulation
of the electrical input. We show that this behavior stems from a perturbation
in the bistable dynamics of the ring laser. In addition to switching and
bistability, our novel coupler design for quantum cascade ring lasers offers
an efficient mechanism for outcoupling and light detection.

##  1 Introduction

Ring lasers have been and continue to be investigated for their rich dynamics
in a variety of geometries and material platforms. Numerous phenomena based on
ring lasers have been numerically studied and experimentally observed, such as
unidirectional and bidirectional operation, alternate oscillations,
bistability and chaos [1, 2, 3, 4, 5, 6]. These phenomena result from the
nonlinear interaction of the clockwise (CW) and counterclockwise (CCW) fields.
These two fields, which in an ideal resonator do not couple to each other,
give rise to the travelling-wave nature of the total field in a ring laser.
This is in contrast to Fabry-Perot (FP) lasers, where the electric field
exhibits a standing-wave pattern leading to spatial hole burning (SHB) of the
gain medium. In lasers where the gain recovery time is fast, such as quantum
cascade lasers (QCLs), SHB effects are dominant and lead to multimode
instabilities [7]. Although ring resonators seem to evade SHB, pure travelling
waves are only possible in an ideal cavity; in a realistic cavity, even small
imperfections will perturb the rotational invariance of the counter-
propagating modes and pin them to a standing-wave pattern similar to FP
lasers. Thus, the only realistic way to obtain travelling waves in a ring
laser is by ensuring unidirectional operation where only one of the two modes,
CW or CCW, is allowed to lase at a time. Unidirectional operation of ring QCLs
has been shown to be necessary for the demonstration of active mode locking
and mid-infrared, ring-based frequency combs [8, 9]. Previous work on QCLs
aimed directly at mode selection between CW and CCW modes has relied on an
"S-shaped" waveguide design to favor one mode [10] or external cavities where
the gain medium is placed within a free-space-coupled ring cavity [8, 11].

In this work, we show robust and deterministic unidirectional operation from a
monolithic racetrack QCL. Instead of a specialized waveguide design of the
ring laser itself, which permanently favors one mode, we rely on an active
waveguide evanescently coupled to the racetrack to perform mode selection and
switching, which allows for controllable operation in both directions. This in
turn, may enable denser integration and greater functionality of mid-infrared
photonic integrated circuits.

Switching between the two modes of a semiconductor ring laser has also been
used to demonstrate random number generation [5]. The fast carrier lifetime of
QCLs ($\sim$ ps) may be used in such applications to improve the speed of this
operation. In addition, we show that our ring laser exhibits a region of
bistability, a dynamical regime that has not yet been explored in mid-infrared
QCLs. The bistable nature of the ring allows for full switching between the
two modes with small input perturbations.

Our QCL operates at $\lambda\approx$ $8\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, a wavelength where the fabrication of an evanescent coupler becomes
challenging because the active region is buried deep in the QC layers thus
requiring a complex etching process. Unlike previous work, which focuses on
short wavelengths ($\lambda<$ $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$), [10,
12] we fabricate an evanescent coupler that does not rely on any specialized
growth or processing techniques.

Lastly, ring QCLs have recently been studied as emerging platforms for
frequency combs [9, 13, 14]. In these studies, light extraction is done by
sensing the weak scattering off the waveguides due to bending losses. While
this technique aids in minimizing backscattering between the two counter-
propagating modes, it may pose a limitation on the practical use of these
devices. In addition to controllable mode selection and switching, the active
waveguide in our system also serves as an efficient outcoupling mechanism for
the racetrack laser.

##  2 Design and Fabrication

We have fabricated a racetrack resonator coupled to a bus waveguide on a
conventional $\lambda\approx$ $8.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ QCL
wafer based on a double-phonon resonance design [15]. The active bus waveguide
differs from the more commonly utilized passive bus waveguide in two key ways.
First, it has intersubband transitions at the same wavelength that the
racetrack emits; thus, to overcome absorption, the waveguide must be pumped to
transparency. Second, due to the same transitions and depending on the pumping
level, it has its own spontaneous emission which is coupled into the racetrack
and is responsible for the switching properties we will discuss here.

Figure 1: Device schematic showing the racetrack laser ridge and the
evanescently-coupled waveguide with facets tilted at $17^{\circ}$. The dotted
purple box shows a magnified view of the taper and s-bend employed on the
ridges in order to enhance their coupling. The blue and red arrows illustrate
how light is coupled in and out of the racetrack. Yellow regions represent a
schematic of the top and ground metal contact pads. The bottom left panel
shows the calculated mode intensity of a $5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ ridge waveguide. The bottom right panel offers a few combinations of gap
width and coupling length leading to particular coupling strengths as
calculated by CMT. The devices discussed here correspond to the last set of
listed parameters.

The schematic of our system is shown in Fig. 1. The two arms of the waveguide
have separate metal contacts and are thus controlled independently. The
coupling mechanism in this system is evanescent coupling where a small gap is
deep etched between the ridges. To determine the size of the gap as well as
the length of the coupler we use coupled mode theory (CMT) simulations based
on the computed fundamental mode. Our CMT simulations show that significant
coupling can only occur for relatively thin ridges. In our case we choose a
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ ridge width because the mode is less
confined laterally thereby expanding outside the waveguide core and enhancing
evanescent coupling. For ease of fabrication in the following processing
steps, we adiabatically taper our waveguide width to
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ outside of the coupling region.
This also helps to limit loss due to weak confinement and proximity of the
mode to the lossy metal on the sidewalls. The detailed design of the coupler
is shown in Fig. 1. We have considered several combinations of gap and
coupling length and their corresponding coupling strength is given in the
table in Fig. 1.

Figure 2: Scanning electron microscope (SEM) images of the device, focusing on
the coupler. Top left: A top-view of the coupling region and taper. Top right:
Cross-sectional view along the taper. The darker contour along the ridges is
the insulating SiNx layer. The coupling region is kept free of metal. Bottom
left: Cross-sectional view of the ridge waveguide elsewhere in the device with
a dielectric window and Ti/Au metal layer for electrical contact. Bottom
right: Magnified view showing the etch profile and minimal, highly sub-
wavelength roughness on the waveguide sidewalls.

The chosen physical dimensions are compatible with conventional lithography
techniques. The most challenging step in the fabrication is deep etching of
the gap with an aspect ratio $\geq 5$ while still maintaining vertical
sidewalls and minimal roughness. We utilize a multistep inductively coupled
plasma – reactive ion etching (ICP-RIE) process based on Cl2 and BCl3. The
ICP-RIE process consists of many parameters, which we optimize for minimal
sidewall roughness. Fig. 2 shows scanning electron microscope (SEM) images of
various elements of the device to especially showcase the etch
characteristics. To reduce bending losses, all bend radii are designed to be
larger than $450\text{\,}\mathrm{\SIUnitSymbolMicro m}$. To avoid feedback
from reflection, the waveguide arms are angled $17^{\circ}$ with respect to
the cleaved facet which lowers their reflectivity to $\approx 0.01$ [16, 17].
We route the waveguides to the same side of the chip for ease of measurement,
while ensuring both arms have practically the same path length to match their
optical properties. Finally, an important design consideration when working
with a system of two or more elements on the same wafer is electrical
crosstalk due to a non-negligible ground impedance. To mitigate this issue,
instead of the more conventional common, substrate-side ground used in QCLs,
we use separate top ground pads for the racetrack laser and each waveguide
arm.

##  3 Temporal Dynamics and Mode Switching

The racetrack laser supports two counter-propagating modes which in an ideal
cavity are best described by travelling waves. Although we optimize the
fabrication process, even minimal sidewall roughness will lead to
backscattering and coupling between the modes. One way to preserve travelling
waves in the cavity is if only one of the counter-propagating modes is allowed
to lase at a time. A mechanism which allows us to select the dominant mode
accurately and to efficiently switch between them is a useful tool to
achieving controllable unidirectional lasing in the racetrack. To distinguish
the two modes experimentally, in our measurements we use a thermoelectrically
(TE)-cooled HgCdTe (MCT) detector with a 1x$1\text{\,}\mathrm{mm}$ active area
along with a pair of ZnSe lenses of 1.5" focal length. By translating the
detector across the length of the device, we are able to fully resolve the two
peaks emitted from both waveguide arms, which correspond to the CW and CCW
modes. All measurements are performed in pulsed mode using
$300\text{\,}\mathrm{ns}$ pulses with $50\text{\,}\mathrm{kHz}$ repetition
rate. The device is thermally stabilized at room temperature with a TE cooler.

We are able to detect the signal from the racetrack laser exiting the
waveguide facets at ports 3 and 4 (shown in the top schematic of Fig. 3) even
without any pump applied to the waveguide arms, although the signal is
attenuated due to significant absorption along each arm. The racetrack
threshold current is $\approx$ $1.25\text{\,}\mathrm{A}$ or
$2.89\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}{\mathrm{c}\mathrm{m}}^{2}$. Our
measurements show that the racetrack laser is unidirectional; however, at each
current pulse, it selects between the CW and CCW mode randomly. Under pulsed
measurements, the output thus fluctuates between the two modes.

We show that a small amount of amplified spontaneous emission injected into
one of the two arms favors one of the modes. In our experimental demonstration
of mode selection we differentially pump the two waveguide arms taking
advantage of the fact that spontaneous light emitted by each arm only couples
to one of the two possible modes. Similarly, each mode will couple to a
different arm of the active waveguide which allows us to distinguish them with
our MCT detector. The schematic in Fig. 3 illustrates these mode interactions
with color-coded arrows. In our experiment we pump the two waveguide arms
differentially by applying a variable current $I=I_{0}\pm\Delta I$ to each
arm. This way the average applied current, $I_{0}$, at any point during our
sweep remains constant. With the external MCT detector we then capture the
signal at ports 3 and 4 (marked in orange in the schematic). The racetrack
signal exiting the facets at 3 and 4 has interacted with the waveguide arms
which provide pump-dependent absorption or amplification. An experimental
measurement of this interaction is shown on the bottom right panel of Fig. 3
which depicts the CCW output at port 3 as a function of the current applied on
the left waveguide arm in the same schematic (which favors CCW operation). For
up to $\approx$ 400 mA of applied current, absorption is the dominant effect;
however, beyond this point, gain in the waveguide takes over and the output is
amplified with increasing current. Measuring this characteristic gain curve
allows us to calculate the output right as it exits the racetrack
(schematically depicted by the turquoise ports 1 and 2). We use a Softplus
function to fit the gain curve and then use this fit to calculate what the
measured signal exiting ports 3 and 4 looks like at ports 1 and 2. This allows
us to investigate and display the mode selection mechanism more clearly
without the additional effects of absorption or amplification in the
waveguide.

Figure 3: Top: System schematic. Bottom left: CW and CCW normalized intensity
given as a function of injected current imbalance between the two waveguide
arms as illustrated in the system schematic. The CW (CCW) mode shown in dotted
red (blue) dominates for positive (negative) $\Delta I$. This is the signal
right as it exits the racetrack, represented by the turquoise markers at ports
1 and 2 in the schematic. In solid red (blue) we show the measured probability
that the mode is CW (CCW) from 15 consecutive measurements. Bottom right: CCW
power measured by an MCT photodetector as it exits the waveguide (noted by an
orange marker at ports 3) along with a Softplus fit.

We have shown on the bottom left panel of Fig. 3 the measured CW and CCW
signals corrected using the fit discussed above as a function of the current
differential between the two waveguide arms. This is the output immediately
after it is outcoupled from the racetrack before it travels through the length
of the waveguide arms. The lasing mode (CW or CCW) is only dependent on which
arm has a higher pump. Following notation in the schematic of Fig. 3, for
$\Delta I>0$, the right arm has a higher spontaneous emission, favoring the CW
mode. Conversely, for $\Delta I<0$, the left arm’s stronger emission favors
the CCW mode. For a small region around $\Delta I=0$, the mode is chosen at
random. In the same plot we show the probability that the mode is CW (red,
solid) or CCW (blue, solid) as computed from a representative set of 15
consecutive measurements for any given applied current. The intensity of the
dominant mode remains relatively constant as we sweep $\Delta I$. Any light
injected into the racetrack from the waveguide arms in these measurements is
due to spontaneous emission. In fact, the two waveguide arms form a FP cavity
and therefore can achieve lasing if pumped together. However, in any of these
measurements, the bus waveguide is well below lasing because we only pump one
arm at a time which leads to significant loss in the bus cavity. Together with
the $17^{\circ}$ facets, this loss leads to a high lasing threshold ($\approx$
$7.84\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}{\mathrm{c}\mathrm{m}}^{2}$ or
$3.36\text{\,}\mathrm{A}$ if both arms are pumped simultaneously).

A mode selection mechanism based on spontaneous emission injection is
consistent with our measurements. We have also considered the role that
feedback of the racetrack signal from the facets back into the racetrack may
play in our system; however, we have found this effect to be weak considering
the combined effect of the weak reflection from the facets and a double pass
through the $15\%$ coupler.

Having shown that we are able to select one of the two modes based on the
current differential $\Delta I$, we now examine the effectiveness of switching
between modes. We measure the amount of current imbalance needed to switch for
various average currents $I_{0}$ applied to the waveguide arms. To this end,
we assess the probability for each mode to dominate by capturing a
representative train of 15 consecutive measurements from the MCT detector as
shown schematically in Fig. 4. We only observe two possible responses from the
photodetector. For example, if we are observing at the CW port, we see a high
photodetector response if the mode is CW or zero response if the mode is CCW
and never in between. The probability $P_{CW(CCW)}$ is then defined as the
number of pulses measured in CW (CCW) mode divided by the total number of
pulses, in our case 15. The results of this analysis are shown in Fig. 4 for
three different values of $I_{0}$. Note the crossover happens around $\Delta
I\approx$0\text{\,}\mathrm{mA}$$. It is always slightly shifted to one side,
which we explain based on a slight imbalance in the lengths of the two
waveguide arms. Another key detail is the span of the bistable region (where
probabilities of each mode are comparable). To determine this span we fit our
data using the error function and calculate the width going from $10\%$ to
$90\%$ probability. For $I_{0}$ = $620\text{\,}\mathrm{mA}$ this region spans
$\approx$ $73.2\text{\,}\mathrm{mA}$. As $I_{0}$ increases to
$900\text{\,}\mathrm{mA}$, the span of this region decreases to
$13.8\text{\,}\mathrm{mA}$.

Figure 4: Top: Schematic of a received pulse train used to measure the
probability of each mode appearing. In this example, the probability
$P_{CW}=P_{CCW}=0.5$. Bottom: Probability of mode selection for the CW (red)
and CCW (blue) modes. The solid line is an error function fit of the raw data
(displayed as dots). The mean current increases from left to right from
$620\text{\,}\mathrm{mA}$ to $900\text{\,}\mathrm{mA}$. Required switching
current, calculated as the width of the $10\%$-$90\%$ interval of each fit,
decreases from $73.2\text{\,}\mathrm{mA}$ to $13.8\text{\,}\mathrm{mA}$ from
left to right. The racetrack current in these measurements is held constant at
$1.81\text{\,}\mathrm{A}$
($4.19\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}{\mathrm{c}\mathrm{m}}^{2}$).

##  4 Bistability Mechanism

Thus far we have shown that we can select the direction in which the racetrack
laser emits by modulating the current differential into the waveguide arms.
However, a question arises on the mechanism that enables this phenomenon. Is
it necessary to continuously pump a particular arm of the waveguide if we wish
to maintain the directionality of the racetrack emission or is it sustained by
the racetrack dynamics itself? To answer this question we conduct the
experiment shown in Fig. 5. The setup is described by the upper panels of the
figure. A $300\text{\,}\mathrm{ns}$ pulse of $\approx$
$1.81\text{\,}\mathrm{A}$ is applied to the racetrack. At this current, the
racetrack is 1.4 times above the lasing threshold. Midway through the
racetrack pulse, we apply a short current pulse of width $\approx$
$30\text{\,}\mathrm{ns}$ and current up to $1.15\text{\,}\mathrm{A}$ to the
waveguide arm that favors CCW operation. In the opposite arm we apply a
$300\text{\,}\mathrm{ns}$ pulse of $\approx$ $480\text{\,}\mathrm{mA}$ just
above the necessary current input for absorption mitigation. Observing the
photodetector response during each pulse before and after the short
$30\text{\,}\mathrm{ns}$ pulse input is one way to address the question we
have posed above. On the lower left plot we show the output on the CCW side
(port 3) measured on a TE-cooled MCT detector when the input CCW spike peak
(in blue) is lower than the longer CW pulse (in red). We plot 15 consecutive
measurements which reveal a seemingly random selection of the lasing
direction. When the laser’s CCW mode dominates, we measure a pulsed signal on
the detector; conversely, when the CW mode dominates, no signal is measured,
as shown in Fig. 5. On the lower-right plot we show photodetector measurements
when the spike injected into the arm favoring CCW operation is higher than the
CW background injected into the other arm. We observe that this input
perturbation causes the racetrack laser to switch to the CCW mode
deterministically, regardless of the initial lasing mode. This can be seen for
the section of the pulse highlighted in gray, where all of the measurements
show a high signal on the detector, corresponding to the CCW mode. Moreover,
the laser remains fixed at the CCW mode, even after the perturbation that
caused the switch dissipates. This sheds some light on the mode selection
mechanism. While pumping the external waveguide arm favors one particular
racetrack mode to dominate, the selected mode is self-sustaining thereafter.
This demonstrates the bistable nature of the racetrack dynamics which allows
for unidirectional mode selection.

Figure 5: Top: Input pulses into the racetrack and each waveguide arm. (a) The
spike coupling to the CCW mode is smaller than the CW background. (b) The
spike coupling to the CCW mode is larger than the CW background. (c), (d) 15
superimposed photodetector traces at port 3 corresponding respectively to
pumping scenarios in (a), (b). Note the difference in the region shaded in
gray between plots (c) and (d). When the short pulse exceeds pumping in the
opposite arm, the CCW mode always dominates. (e) Device schematic along with
mode interactions.

##  5 Spectral Characteristics

Figure 6: Spectral characteristics of the device. The left panel corresponds
to the spectrum of the racetrack when no current is applied to the waveguide
arms. For the other panels, the average applied current $I_{0}$ is given at
the top and the spectra are given as a function of $\Delta I$ where negative
(positive) $\Delta I$ favors the CW (CCW) mode. The spectra in each panel are
normalized to the maximum intensity point of that panel.

The spectral properties of the racetrack laser and the impact that pumping of
the waveguide arms has on them are summarized in Fig. 6. We measure the signal
exiting one of the waveguide arms using a Fourier Transform Infrared
Spectrometer (FTIR) with $0.125\text{\,}{\mathrm{c}\mathrm{m}}^{-1}$ spectral
resolution. The pure racetrack spectrum with no current applied to either
waveguide arm is shown in the left panel and it reveals a series of
longitudinal modes spanning $\approx$ $14\text{\,}{\mathrm{c}\mathrm{m}}^{-1}$
with a measured free-spectral range (FSR) of $\approx$
$0.70\text{\,}\mathrm{c}\mathrm{m}^{-1}$ which is an excellent match for the
calculated value $1/nL$ = $0.701\text{\,}\mathrm{c}\mathrm{m}^{-1}$, where
$L=$4.318\text{\,}\mathrm{mm}$$ is the designed racetrack cavity length and
$n=3.3$ is the effective refractive index of the ridge waveguide. On the right
panel of Fig. 6 we show the CW spectra as a function of current imbalance
between the arms $\Delta I$ as well as average current $I_{0}$ which is marked
at the top. In this case, a positive $\Delta I$ means higher current is
applied to the waveguide arm favoring CCW operation which explains why the
spectra within each panel are generally weaker for $\Delta I>0$.

The FSR, once some current is applied to the waveguide arms, remains the same
as for the pure racetrack spectrum, although there is a small red shift in the
center frequency which can be explained by increased local temperature of the
chip due to the applied current $2I_{0}$. For low values of $I_{0}$, despite
the red shift, the spectral envelope that we measure remains similar to the
racetrack spectrum without injected current. As $I_{0}$ increases this
envelope changes significantly with only a few modes becoming dominant as can
be seen for $I_{0}=$ $790\text{\,}\mathrm{mA}$ and $I_{0}=$
$900\text{\,}\mathrm{mA}$. For these high values of $I_{0}$ we are able to
recover the pure racetrack spectrum around $\Delta I=0$ mark. This is most
visible for $I_{0}=$ $790\text{\,}\mathrm{mA}$ where even though only a couple
of modes dominate for negative $\Delta I$, around the 0 mark, we recover the
original envelope. Hence, lower values of current applied to the waveguide
arms do not alter the spectral envelope of the racetrack laser; however, for
higher values the envelope changes significantly. When the current
differential between the arms is relatively small, the resulting spectra are
also similar to the pure racetrack spectrum envelope.

##  6 Conclusion

In this paper we have shown the operation of a novel monolithic QCL-based
system comprising of a racetrack laser and an evanescently-coupled bus
waveguide with independently-controlled arms based on a specially-designed
tapered coupler. We demonstrate deterministic and controllable selection
between the two counter-propagating racetrack modes via spontaneous emission
coupling from the differentially pumped waveguide arms. As little as
$14\text{\,}\mathrm{mA}$ of current imbalance, which represents $\approx
1.6\%$ of the average input current into the waveguides, can trigger a full
and permanenet switch between the CW and CCW modes. By further investigating
the mechanism of mode selection, we have demonstrated the possibility of
bistable operation of our device. Because of the bistable nature of the ring,
the waveguide arms do not need constant pumping in order to maintain a
particular directional mode. In fact, a small perturbation is sufficient to
select the mode and the racetrack laser will maintain this mode even after the
input perturbation dissipates.

Finally, we have demonstrated an efficient outcoupling mechanism by measuring
up to $\approx$ $4\text{\,}\mathrm{mA}$ of optical power coupled out of the
racetrack laser. These results bring us closer towards practical applications
of controllable unidirectional ring or racetrack QCLs.

### Funding

This work was supported by the Eric and Wendy Schmidt Transformative
Technology Fund. S.K. acknowledges support from the Yan Huo *94 Graduate
Fellowship.

### Acknowledgments

We acknowledge the Micro and Nano Fabrication Center (MNFC) of Princeton
University where fabrication was carried out.

### Disclosures

The authors declare no conflicts of interest.

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# Learning Proxemic Behavior Using Reinforcement Learning with Cognitive
Agents

Cristian Millán-Arias1, Bruno Fernandes1, Francisco Cruz2,3 3Escuela de
Ingeniería, Universidad Central de Chile, Santiago, Chile
Emails: {ccma<EMAIL_ADDRESS><EMAIL_ADDRESS>2School of
Information Technology, Deakin University, Geelong, Australia
1Escola Politécnica de Pernambuco, Universidade de Pernambuco, Recife-PE,
Brazil

###### Abstract

Proxemics is a branch of non-verbal communication concerned with studying the
spatial behavior of people and animals. This behavior is an essential part of
the communication process due to delimit the acceptable distance to interact
with another being. With increasing research on human-agent interaction, new
alternatives are needed that allow optimal communication, avoiding agents
feeling uncomfortable. Several works consider proxemic behavior with cognitive
agents, where human-robot interaction techniques and machine learning are
implemented. However, environments consider fixed personal space and that the
agent previously knows it. In this work, we aim to study how agents behave in
environments based on proxemic behavior, and propose a modified gridworld to
that aim. This environment considers an issuer with proxemic behavior that
provides a disagreement signal to the agent. Our results show that the
learning agent can identify the proxemic space when the issuer gives feedback
about agent performance.

###### Index Terms:

Cognitive Agents, Proxemics, Reinforcement Learning

##  I Introduction

Proxemics is the study of spatial behavior, concerned with territoriality,
interpersonal distance, spatial arrangements, crowding, and others aspects of
the physical environment that affect behavior. The term was coined by Hall et
al. [1], when he proposes a fixed measure of personal space, a set of regions
around a person to delimit the acceptable distance to interact with other
people. In recent years, human-agent interaction has taken hold in the
scientific community. Furthermore, the humanization of agents is an expected
event, given technological advances and human nature. Thus, an optimal
interaction is necessary on both the agent and the person [2].

Reinforcement Learning (RL) is a learning paradigm that tries to solve the
problem of an agent interacting with the environment to learn the desired task
autonomously [3]. The agent must sense a state from the environment and take
actions that affect it to reach a new state. The agent receives a reward
signal from the environment that it tries to maximize throughout the learning
for each action taken. The agent takes actions from its own experience, or can
be guided by an external trainer that provides feedback [4, 5]. Proxemic
behavior has been used in different areas with cognitive agents. For example,
in human-robot interaction, it has been studied how people behave in the
presence of an artificial agent or robot and how their perception of the
personal space of the agent has been [6, 7]. Moreover, in machine learning, it
has been studied how artificial agents sense the personal space of other
cognitive agents, and it has been to identify and learn personal space [8, 9].

##  II Proxemic Behavior in Cognitive Agents

We propose a modified version of the gridworld problem. In this environment,
an issuer is placed on one fixed state and is responsible for giving a signal
of disagreement when the learning agent is too close. Two regions are defined
around the issuer, the uncomfortable region and the target region. The
uncomfortable region is those states that make up a square around the issuer.
In this work, the uncomfortable region is only an area of negative reward.
Similarly, the target region is those states that make up a square around the
uncomfortable region. A new action, PING, is added to the four traditional
ones (UP, DOWN, LEFT, RIGHT). This action represents a communication signal
with the issuer, i.e., the agent send a ping to ask the issuer if it is in the
target region. The task finishes in three conditions:

  * •

Condition C1: When the agent reaches the issuer.

  * •

Condition C2: When the agent sends ping five times out of the target region.

  * •

Condition C3: When the agent sends ping in the target region.

The reward function is defined as:

$\rho=\rho_{issuer}+\rho_{grid},$ (1)

where $\rho_{issuer}$ is a numerical reward given by the issuer when the agent
performs the PING action, and $\rho_{grid}$ is the environment reward defined
as:

$\rho_{grid}=\left\\{\begin{array}[]{rcl}-1.0&\mbox{if}&\mbox{conditions C1 or
C2}\\\ -0.8&\mbox{if}&\mbox{reach to uncomfortable region}\\\
-0.4&\mbox{if}&\mbox{give incorrectly ping}\\\ -0.1&\mbox{if}&\mbox{another
state}\\\ +1.0&\mbox{if}&\mbox{condition C3}\\\ \end{array}\right..$ (2)

In our experiments, we use a $10\times 12$ gridworld, the issuer is placed in
$(6,8)$ and maintains fixed during the training. Each agent starts the
training in the superior left corner of the grid $(0,0)$.

##  III experimental results

To explore the behavior of agents in a proxemic environment, we apply the
Q-Learning algorithm [10] in the modified gridworld problem. We use
$\epsilon$-greedy to select random actions. We set the values of $\epsilon$
and learning rate $\alpha$ at 0.6 and the discount factor $\gamma$ at 0.9. In
our experiments, $100$ agents are trained with $10000$ time-steps.

In our experiments, we consider three scenarios:

  * •

Scenario S1: When $\rho_{issuer}=0$ in each PING action selection.

  * •

Scenario S2: When $\rho_{issuer}$ is a random value in $(-1,1)$.

  * •

Scenario S3: When
$\rho_{issuer}=-\dfrac{d_{1}(P_{agent},P_{issuer})}{d_{1}(P_{start},P_{issuer})}$,
where $d_{1}(\cdot,\cdot)$ is the $L_{1}$ distance, $P_{agent}$, and
$P_{issuer}$, and $P_{start}$ are the position on the grid from the agent, the
issuer and the start point. This value indicates that states farthest from the
issuer have a low reward, and the denominator standardizes it on (-1, 0).

(a) Scenario S1 (b) Scenario S2 (c) Scenario S3

(d) Scenario S1 (e) Scenario S2 (f) Scenario S3

(g) Scenario S1 (h) Scenario S2 (i) Scenario S3

Figure 1: Results of our experiments in three scenarios. The first row
(1(a)-1(c)) shows the average of maximum Q values per time step, the shaded
area is between the maximum and minimum Q values from 100 agents. The second
row (1(d))-1(f)) shows the average of final Q value for the actions UP, DOWN,
LEFT, and RIGHT. The last row (1(g)-1(i)) show the average final Q values for
the PING action.

Fig. 1 shows the results of our experiments in the three scenarios. In terms
of maximum Q-value per time step (Fig. 1(a)-1(c)), there exists more
variability when the issuer gives random rewards. However, according to the
other scenarios, the maximum Q-value reaches values around $1$ in several time
steps. On the other hand, when the issuer gives rewards based on distance, the
variability of Q-values is minor.

The Q values for each action are more focused on the target region than in the
other scenarios when the issuer gives rewards based on distance (Fig.
1(d)-1(f)). This is expected due to the agent receives additional information
on how to move through the grid.

Concerning the PING, the issuer reward highlights the importance of giving
ping in the target region, as shown in Fig. 1(g)-1(i). The graph shows how the
states of the target region have only Q-values greater than zero. While with
the lack of information (Scenario S1), states outside the target region have
Q-values greater than zero.

##  IV Conclusions and future alignments

In this paper, we study the agent performance in an environment based on
proxemic behavior. We implement a modified gridworld problem, where an issuer
agent gives a signal of disagreement when the RL agent performs a ping action.
Moreover, the target is a proxemic region around the issuer instead of the
traditional goal state. In our experiments, we consider three scenarios based
on the reward type that the issuer gives. Our results show that the agent can
reach the target region, even when the issuer does not give information. On
the other hand, the issuer reward gives more information about performing the
ping action, even when random information is given. Thus, the agent can
identify the correct region to give a ping signal.

In our implementation, we consider that the proxemic region is symmetric
around the user. Thus, in our future alignments, we intend to implement
asymmetric proxemic regions to mimic the human proxemic behavior. Also, we
contemplate implementing no fixed proxemic region, considering that the
proxemic space changes by external factors. Finally, we intend to study the
agent performance in more complex environments, involving other algorithms and
techniques of reinforcement learning and deep learning.

## Acknowledgment

We would like to gratefully acknowledge financing in part by the Coordenacão
de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES) - Finance Code
001, and the Brazilian agencies FACEPE and CNPq - Code 432818$/$2018-9.

## References

  * [1] E. T. Hall, R. L. Birdwhistell, B. Bock, P. Bohannan, A. R. Diebold Jr, M. Durbin, M. S. Edmonson, J. Fischer, D. Hymes, S. T. Kimball _et al._ , “Proxemics [and comments and replies],” _Current anthropology_ , vol. 9, no. 2/3, pp. 83–108, 1968.
  * [2] N. Churamani, F. Cruz, S. Griffiths, and P. Barros, “icub: learning emotion expressions using human reward,” _arXiv preprint arXiv:2003.13483_ , 2020\.
  * [3] R. S. Sutton and A. G. Barto, _Reinforcement Learning: An Introduction_. Cambridge, Massachusetts: MIT press, 2018.
  * [4] C. Millán, B. Fernandes, and F. Cruz, “Human feedback in continuous actor-critic reinforcement learning,” in _Proceedings European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning_ , no. April, Bruges (Belgium), 2019, pp. 661–666.
  * [5] C. Millan-Arias, B. Fernandes, F. Cruz, R. Dazeley, and S. Fernandes, “A robust approach for continuous interactive actor-critic algorithms,” _IEEE Access_ , vol. 9, pp. 104 242–104 260, 2021.
  * [6] J. Mumm and B. Mutlu, “Human-robot proxemics: physical and psychological distancing in human-robot interaction,” in _Proceedings of the 6th international conference on Human-robot interaction_ , 2011, pp. 331–338.
  * [7] G. Eresha, M. Häring, B. Endrass, E. André, and M. Obaid, “Investigating the influence of culture on proxemic behaviors for humanoid robots,” in _2013 IEEE Ro-Man_. IEEE, 2013, pp. 430–435.
  * [8] P. Patompak, S. Jeong, I. Nilkhamhang, and N. Y. Chong, “Learning proxemics for personalized human–robot social interaction,” _International Journal of Social Robotics_ , vol. 12, no. 1, pp. 267–280, 2020.
  * [9] N. Mitsunaga, C. Smith, T. Kanda, H. Ishiguro, and N. Hagita, “Robot behavior adaptation for human-robot interaction based on policy gradient reinforcement learning,” _Journal of the Robotics Society of Japan_ , vol. 24, no. 7, pp. 820–829, 2006.
  * [10] C. J. Watkins and P. Dayan, “Q-learning,” _Machine learning_ , vol. 8, no. 3-4, pp. 279–292, 1992.


#  A KURATOWSKI THEOREM REVISITED

Sanjib Basu Department of Mathematics,Bethune College,181 Bidhan Sarani
<EMAIL_ADDRESS>and  Abhit Chandra Pramanik Department of Pure
Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata
700019, West Bengal, India<EMAIL_ADDRESS>
###### Abstract.

If $f:X\mapsto Y$ is a function having Baire property from a metric space $X$
into a separable metric space $Y$, then $f$ is continuous except on a set of
first category. Kuratowski asked whether the condition of separability could
be removed. Several attempts to solve this problem were done in the past, the
earliest one by Kinjiro Kunuji in 1936 based on a theorem of Luzin and
Novikov. Recently, Grzegorek and Labuda proved Kunuji’s theorem based on a
generalization of an earlier result of Kuratowski in second countable
topological space. In this paper, we develope an unification of this
generalization of Kuratowski’s result and its measure theoretic analogue in
category bases and utilize the same to give a generalized version of Kunuji’s
theorem. A variant of this unification is also obtained in Marczewski
structures.

###### Key words and phrases:

Category bases, meager sets, non-Baire sets, pseudo base, Marczewski base,
$s_{0}$-sets, $s$-sets, Montegomery lemma

###### 2020 Mathematics Subject Classification:

28A05, 54A05, 54E52

The second author thanks the CSIR, New Delhi – 110001, India, for financial
support

##  1\. INTRODUCTION

In a paper, published in 1976 [5], Kuratowski proved the following theorem :

######  Theorem 1.1.

Let $Y$ be a complete separable metric space equipped with a non-trivial
finite measure $\mu$. Let {$A_{\gamma}:\gamma\in\Gamma$} be a partition of $Y$
into first category (resp, $\mu$-measure zero) sets. If the continuum
hypothesis holds, then there exists $\Delta\subseteq\Gamma$ such that
$\bigcup\limits_{\gamma\in\Delta}A_{\gamma}$ does not have the Baire property
(resp, is not $\mu$-measurable).

The measure part of the above result was already known to Solovay. In 1970, he
proved (an unpublished result) using generic ultrafilters that any partition
of [0,1] into Lebesgue measure zero sets produces a non-measurable set (for
reference, see [7]). In 1979, Bukovsky [1] using the same forcing argument
gave a much shorter and less complicated proof of Solovay’s result. In the
same issue of the journal, there appeared another paper by Emeryk, Frankiewicz
and Kulpa [2] which provided a topologcal proof of the ‘Baire part’ of
Kuratowski’s theorem in Ćech complete spaces. The result is as follows :

######  Theorem 1.2.

Let $X$ be a Ćech complete space of weight $\leq 2^{\omega}$ and $\mathcal{F}$
be a partition of $X$ into meager sets. Then there exists a subfamily
$\mathcal{A}\subseteq\mathcal{F}$ such that $\bigcup\mathcal{A}$ has not the
Baire property.

The above attempts to establish an improved version of Kuratowski’s result
were all directed towards solving a 1935 problem of Kuratowski. It was already
known that if $X$ is a metric space, $Y$ is a separable metric space and
$f:X\mapsto Y$ is a function having Baire property, then $f$ is continuous on
the complement of a set of first category. Kuratowski [6] (1935) asked the
question whether the same is true if $Y$ is not separable. To this, Emeryk’s
[3] answer (which also appeared in 1979) and which uses the results of Emeryk,
Frankiewicz and Kulpa mentioned above, is as follows :

######  Theorem 1.3.

Let $X$ be a C̀ech complete metric space of weight less than or equal to
$2^{\aleph_{0}}$. If $f:X\mapsto Y$ is a function having Baire property from
$X$ into a metric space $Y$, then $f$ is continuous on the complement of a set
of first category.

Solovay also produced an answer (as quoted in [7]) to the measure analogue of
the above problem in the following form :

######  Theorem 1.4.

If $X$ is a metric space and $f:[0,1]\mapsto X$ is measurable, then $f$ is
almost continuous (Luzin measurable).

Only recently, it came to light that an attempt to solve the Kuratowski
problem actually originated much earlier in a paper of 1936 by Kunuji [8] (see
also [4]). Kunuji, who was also after the problem of Kuratowski, imposed the
following restriction on the space $X-$

“ If given an ordinal number $\gamma$, {$A_{\gamma}\\}_{\xi<\gamma}$ is a
family of mutually disjoint first category sets such that their union
$\bigcup\limits_{\xi<\gamma}A_{\xi}$ is a set of second category, then there
exist an open set $O$ and two disjoint unions
$\bigcup\limits_{\xi^{\prime}}A_{\xi^{\prime}}$ and
$\bigcup\limits_{\xi^{\prime\prime}}A_{\xi^{\prime\prime}}$ of subfamilies
each of which is everywhere of second category in $O$ ”

and proved that $-$

######  Theorem 1.5.

If $X$ is a topological space satisfying the above restriction and $f:X\mapsto
Y$ is a function having Baire property from $X$ into a metric space $Y$, then
$f$ is continuous except on a set of first category in $X$.

Grzegorek and Labuda [4] have shown that the relevant topological fact which
implies the validity of Kunuji’s assumption is that the space should have a
countable base of open sets, or, in otherwords, it should be second countable.

Below we state the result of Grzegorek and Labuda followed by its measure
analogue.

######  Theorem 1.6.

Let {$A_{\gamma}:\gamma\in\Gamma$} ($\Gamma$ is an index set) be a family of
mutually disjoint first category subsets of a second countable topological
space $X$. If the union $E=\bigcup\limits_{\gamma\in\Gamma}A_{\gamma}$ is a
set of second category, then there exist $\Delta\subseteq\Gamma$ and an open
set $O$ such that both $\bigcup\limits_{\gamma\in\Delta}A_{\gamma}$ and
$\bigcup\limits_{\gamma\in{\Gamma\setminus\Delta}}A_{\gamma}$ are everywhere
of second category in $O$. As a consequence,
$\bigcup\limits_{\gamma\in\Delta}A_{\gamma}$ and
$\bigcup\limits_{\gamma\in{\Gamma\setminus\Delta}}A_{\gamma}$ cannot be
separated by sets having Baire property.

######  Theorem 1.7.

Let {$A_{\gamma}:\gamma\in\Gamma$} be a family of mutually disjoint measure
zero sets in a second countable topological space. If the union
$E=\bigcup\limits_{\gamma\in\Gamma}A_{\gamma}$ is a set of positive outer
measure, then there exists $\Delta\subseteq\Gamma$ such that
$\bigcup\limits_{\gamma\in\Delta}A_{\gamma}$ and
$\bigcup\limits_{\gamma\in{\Gamma\setminus\Delta}}A_{\gamma}$ cannot be
separated by measurable sets.

In this paper, we develope an unification of Theorem 1.6 and Theorem 1.7 in
category bases and utilize the same to give a generalized version of Theorem
1.5 of Kunuji.

##  2\. PRELIMINARIES AND RESULTS

The concept of a category base is a generalization of both measure and
topology. Its main objective is to present both measure and Baire category
(topology) and some other aspects of point set classification within a common
framework. It was introduced by J.C.Morgan II [9] in the seventies of the last
century and since then has been developed through a series of papers [10],
[11], [12], [13] etc.

The relevant definitions are as follows $-$

######  Definition 2.1.

A pair $(X,\mathcal{C}$) where $X$ is a non-empty set and $\mathcal{C}$ is a
family of subsets of $X$ is called a category base if the non-empty members of
$\mathcal{C}$ called regions satisfy the following set of axioms :

  1. (1)

Every point of $X$ is contained in at least one region; i.e., $X=\cup$
$\mathcal{C}$.

  2. (2)

Let $A$ be a region and $\mathcal{D}$ be any non-empty family of disjont
regions having cardinality less than the cardinality of $\mathcal{C}$.
i) If $A\cap(\cup\mathcal{D}$) contains a region, then there exists a region
$D\in\mathcal{D}$ such that $A\cap D$ contains a region.
ii) If $A\cap(\cup\mathcal{D})$ contains no region, then there exists a region
$B\subseteq A$ which is disjoint from every region in $\mathcal{D}$.

######  Definition 2.2.

$[9]$ In a category base ($X,\mathcal{C}$), a set is called ‘singular’ if
every region contains a region which is disjoint from the set itself. A set
which can be expressed as a countable union of singluar sets is called
‘meager’. Otherwise, it is called ‘abundant’. A set is called ‘Baire’ if every
region contains a subregion in which either the set or its complement is
meager.

A category base is called a ‘Baire base’ if every region in it is abundant.

######  Definition 2.3.

$[9]$ In a category base ($X,\mathcal{C}$), a subfamily $\mathcal{B}$ of
regions is called a ‘pseudobase’ if every region contains a region from
$\mathcal{B}$.

In what follows, we assume that our category base is
($\star$) A Baire base with the property that for every region $A$, the
induced category base ($A,\mathcal{C}_{|A}$) has a countable pseudobase where
$\mathcal{C}_{|A}=\\{B\in\mathcal{C}:B\subseteq A\\}$.

Let {$A_{\alpha}:\alpha\in\Lambda\\}$ ($\Lambda$ is an index set) be a family
of disjoint meager sets in ($X,\mathcal{C}$) such that
$\bigcup\limits_{\alpha\in\Lambda}A_{\alpha}$ is abundant. Let $\Omega$ be the
least ordinal representing card $\Lambda$. Without loss in generality, we may
assume that $\Omega$ is also the least ordinal for which
$\bigcup\limits_{\alpha<\Omega}A_{\alpha}$ is abundant.

Now let
$B_{\alpha}=\bigcup\limits_{\beta<\alpha}A_{\beta}=\bigcup\limits_{n=1}^{\infty}B_{\alpha}^{(n)}$
($\alpha<\Omega$), where $B_{\alpha}^{(n)}$ are singular sets in $X$ (this way
of representing of $B_{\alpha}$ is justified according to the choice of the
ordinal $\Omega$). For any $y\in\bigcup\limits_{\alpha<\Omega}A_{\alpha}$, one
can find $\alpha$ such that $y\in B_{\beta}$ for every $\beta>\alpha$. If
$\gamma$ is the least ordinal for which $y\in B_{\gamma}$, then
$\bigcup\limits_{n=1}^{\infty}\mathcal{U}_{n,y}=\bigcup\limits_{\alpha<\Omega}A_{\alpha}-\bigcup\limits_{\alpha<\gamma}A_{\alpha}$
where $\mathcal{U}_{n,y}=\bigcup\\{A_{\alpha}:y\in B_{\alpha}^{(n)}\\}$ which
is abundant in every region in which
$\bigcup\limits_{\alpha<\Omega}A_{\alpha}$ is abundant.

By the Fundamental theorem (Ch 1, II, F, [9]),
$\bigcup\limits_{\alpha<\Omega}A_{\alpha}$ is abundant everywhere in some
region $D$ (say). Consequently, for every
$y\in\bigcup\limits_{\alpha<\Omega}A_{\alpha}$,
$\bigcup\limits_{n=1}^{\infty}\mathcal{U}_{n,y}$ is abundant everywhere in $D$
and if {$D_{n}:n\in\mathbb{N}$} ($\mathbb{N}$ is the set of natural numbers)
constitutes a countable pseudobase for ($D,\mathcal{C}_{|D}$), then for every
$y$ there is a pair ($n_{y},m_{y})\in\mathbb{N}\times\mathbb{N}$ such that
$\mathcal{U}_{n_{y},y}$ is abundant everywhere in $D_{m_{y}}$. We set
$T_{m,n}=\\{y\in\bigcup\limits_{\alpha<\Omega}A_{\alpha}:\mathcal{U}_{n_{y},y}$
is abundant everywhere in $D_{m_{y}};n_{y}=n,m_{y}=m$} so that
$\bigcup\limits_{m,n=1}^{\infty}T_{m,n}$ is abundant everywhere in $D$.
Consequently, there exists a region $P_{m_{0}}$ of $D_{m_{0}}$ and a pair
($n_{0},m_{0})\in\mathbb{N}\times\mathbb{N}$ such that $T_{n_{0},m_{0}}$ is
abundant everywhere in $P_{m_{0}}$. In otherwords,
$T_{n_{0},m_{0}}=\\{y\in\bigcup\limits_{\alpha<\Omega}A_{\alpha}:\bigcup\\{A_{\alpha}:y\in
B_{\alpha}^{(n_{0})}\\}$ is abundant everywhere in $D_{m_{0}}$} is abundant
everywhere in $P_{m_{0}}$.

Let {$P_{k}^{(m_{0})}:k\in\mathbb{N}$} be a countable pseudobase for
($P_{m_{0}},\mathcal{C}_{|P_{m_{0}}}$). We set
$M_{k}=\bigcup\\{A_{\alpha}:B_{\alpha}^{(n_{0})}\cap P_{k}^{(m_{0})}=\phi$}.
Since
$\bigcup\limits_{k=1}^{\infty}M_{k}=\bigcup\limits_{\alpha<\Omega}A_{\alpha}$,
there exists $k_{0}\in\mathbb{N}$ such that $M_{k_{0}}$ is abundant everywhere
in $P_{k_{0}}^{(m_{0})}$. Choose $y\in T_{n_{0},m_{0}}\cap P_{k_{0}}^{m_{0}}$
and label it as $y_{0}$. Then $\bigcup\\{A_{\alpha}:y_{0}\in
B_{\alpha}^{(n_{0})}\\}=\mathcal{U}_{n_{0},y_{0}}$ and $M_{k_{0}}$ which are
obviously disjoint unions are both abundant in $P_{k_{0}}^{(m_{0})}$.

From the above deductions, we can finally conclude that there exists a region
$E$ and a pair ($\Lambda_{1},\Lambda_{2}$) such that
$\Lambda_{1}\cup\Lambda_{2}=\Lambda$, $\Lambda_{1}\cap\Lambda_{2}=\phi$ and
both $\bigcup\limits_{\alpha\in\Lambda_{1}}A_{\alpha}$ and
$\bigcup\limits_{\alpha\in\Lambda_{2}}A_{\alpha}$ are abundant everywhere in
$E$. Hence $\bigcup\limits_{\alpha\in\Lambda_{1}}A_{\alpha}$ (likewise,
$\bigcup\limits_{\alpha\in\Lambda_{2}}A_{\alpha}$) are non-Baire sets in our
base ($X,\mathcal{C}$).

The above deduction leads to the following

######  Theorem 2.4.

Let ($X,\mathcal{C}$) be a Baire base satisfying the condition ($\star$)
stated above. Let {$A_{\alpha}:\alpha\in\Lambda$} be a family of disjoint
meager sets such that their union is abundant in ($X,\mathcal{C}$). Then there
exist two subfamilies {$A_{\alpha}:\alpha\in\Lambda_{1}$} and
{$A_{\alpha}:\alpha\in\Lambda_{2}$} whose union is
$\bigcup\limits_{\alpha\in\Lambda}A_{\alpha}$ and a region $C$ such that both
$\bigcup\limits_{\alpha\in\Lambda_{1}}A_{\alpha}$ and
$\bigcup\limits_{\alpha\in\Lambda_{2}}A_{\alpha}$ are abundant everywhere in
$C$.

Consequently, by (Th 2, III, C, Ch 1, [9]) it follows that $-$

######  Corollary 2.5.

If ($X,\mathcal{C}$) is a Baire base satisfying condition ($\star$) and
{$A_{\alpha}:\alpha\in\Lambda$} is a family of disjoint meager sets such that
their union is abundant in ($X,\mathcal{C}$), then
$\bigcup\limits_{\alpha\in\Lambda_{1}}A_{\alpha}$ and
$\bigcup\limits_{\alpha\in\Lambda_{2}}A_{\alpha}$ cannot separated by Baire
sets. Consequently, none of $\bigcup\limits_{\alpha\in\Lambda_{1}}A_{\alpha}$
and $\bigcup\limits_{\alpha\in\Lambda_{2}}A_{\alpha}$ is a Baire set.

(Here by separating two disjoint sets $A$ and $B$ by Baire sets, we mean that
there exist a Baire set $E$ in ($X,\mathcal{C}$) such that $A\subseteq E$ and
$E\cap B=\phi$)

This unifies Theorem 1.6 and Theorem 1.7 as desired. We now embark on our
formulation of Kunuji’s theorem in this general settings of category bases. To
start with, we assume that the co-domain space $Y$ of our function $f:X\mapsto
Y$ is a topological spaces satisfying the following condition $-$

($\star$)($\star$) $Y=\bigcup\limits_{n,k=1}^{\infty}D_{n,k}$ (where
$D_{n,k}=\bigcup\limits_{\xi}D_{n,k}^{(\xi)}$) is a disjoint union of Borel
sets such that

  1. (i)

The union $\bigcup\limits_{\xi^{\prime}}D_{n,k}^{(\xi^{\prime})}$ of any
subfamily is also a Borel set .

  2. (ii)

Every nonempty open subset $G$ of $Y$ contains some $D_{n,k}^{(\xi)}$, so that
$G$ can be expressed as union of subfamilies {$D_{n,k}^{(\xi^{\prime})}$} of
{$D_{n,k}^{(\xi)}$} for which $D_{n,k}^{(\xi^{\prime})}\subseteq G$. More
precisely, there exists a set $\Sigma\subseteq\mathbb{N}\times\mathbb{N}$ such
that
$G=\bigcup\limits_{(n,k)\in\Sigma}\bigcup\limits_{\xi^{\prime}}D_{n,k}^{(\xi^{\prime})}$

By virtue of Montegomery lemma (in the form it is used in [4]), any non-
separable metric space is a topological space of the above type.

######  Theorem 2.6.

Let ($X,\mathcal{C}$) be a Baire base where nonempty intersection of any two
region is not singular and $f:X\mapsto Y$ is a Baire function from $X$ into a
topological space $Y$ of the above type. Then there exists a meager set $P$
such that the restriction $f_{|X-P}$ has the following property : For every
nonempty open subset $H$ of $Y$,
$f_{|X-P}^{-1}(H)=\bigcup\limits_{n=1}^{\infty}(\bigcup\mathcal{C}_{H}^{(n)})-P$,
where for each $n\in\mathbb{N}$, $\bigcup\mathcal{C}_{H}^{(n)}$ is a disjoint
union of regions.

###### Proof.

For each pair $(n,k)\in\mathbb{N}\times\mathbb{N}$, since
$D_{n,k}=\bigcup\limits_{\xi}D_{n,k}^{(\xi)}$ is a union of Borel sets, so
$f^{-1}(D_{n,k})=\bigcup\limits_{\xi}f^{-1}(D_{n,k}^{(\xi)})=\bigcup\limits_{\xi}(M_{n,k}^{(\xi)}-F_{n,k}^{(\xi)})\cup
T_{n,k}^{(\xi)}$, where $M_{n,k}^{(\xi)}$ is a disjoint union of regions and
$F_{n,k}^{(\xi)}$, $T_{n,k}^{(\xi)}$ ($F_{n,k}^{(\xi)}\subseteq
M_{n,k}^{(\xi)};F_{n,k}^{(\xi)}\cap T_{n,k}^{(\xi)}=\phi$) are meager sets
(this presentation follows from Th 7, III, E, Ch 1, [9]). Moreover, as
$D_{n,k}^{(\xi_{1})}\cap D_{n,k}^{(\xi_{2})}=\phi$ for $\xi_{1}\neq\xi_{2}$
from the hypothesis and the fact that ($X,\mathcal{C}$) is a Baire base, it
follows that $M_{n,k}^{(\xi_{1})}\cap M_{n,k}^{(\xi_{2})}=\phi$.
We now check that $\bigcup\limits_{\xi}F_{n,k}^{(\xi)}$ is also meager. For
this we need only establish that if {$C_{\alpha}\\}_{\alpha\in\Lambda}$ is a
family of disjoint regions and $B_{\alpha}(\subseteq C_{\alpha})$ are meager,
then $\bigcup\limits_{\alpha\in\Lambda}B_{\alpha}$ is also meager. But this is
a consequence of the fact that
$B_{\alpha}=\bigcup\limits_{n=1}^{\infty}B_{\alpha}^{(n)}$ where
$B_{\alpha}^{(n)}$ are singular and therefore
$\bigcup\limits_{\alpha\in\Lambda}B_{\alpha}^{(n)}$ is singular for each $n$
which may be derived from the definition of a category base.
Now
$\bigcup\limits_{\xi}f^{-1}(D_{n,k}^{(\xi)})=\bigcup\limits_{\xi}(M_{n,k}^{(\xi)}-F_{n,k}^{(\xi)})\cup
T_{n,k}^{(\xi)}$ which by (i) in condition ($\star$)($\star$) can be expressed
as ($M_{n,k}-F_{n,k})\cup T_{n,k}$ where $M_{n,k}$ is the union of disjoint
regions and $F_{n,k},T_{n,k}(F_{n,k}\subseteq M_{n,k};T_{n,k}\cap
M_{n,k}=\phi$) are both meager. Further, if we set
$L_{n,k}=\bigcup\limits_{\xi}M_{n,k}^{(\xi)}$, then
$\bigcup\limits_{\xi}T_{n,k}^{(\xi)}\subseteq\bigcup\limits_{\xi}F_{n,k}^{(\xi)}\cup\\{(M_{n,k}-F_{n,k})-L_{n,k}\\}\cup
T_{n,k}$. Let $S_{n,k}=(M_{n,k}-F_{n,k})-L_{n,k}$. In order to show that
$\bigcup\limits_{\xi}T_{n,k}^{(\xi)}$ is meager, we need to establish that
$S_{n,k}$ is meager.
If possible, let $S_{n,k}$ be abundant. Then $\bigcup\limits_{\xi}(S_{n,k}\cap
T_{n,k}^{(\xi)})$ is also abundant expressed as the disjoint union of meager
sets. So by corollary 2.5, there is a subfamily {$T_{n,k}^{(\xi^{\prime})}$}
of {$T_{n,k}^{(\xi)}$} such that $\bigcup\limits_{\xi^{\prime}}(S_{n,k}\cap
T_{n,k}^{(\xi^{\prime})})$ is non-Baire. But this is impossible, beacause
$\bigcup\limits_{\xi^{\prime}}(S_{n,k}\cap
T_{n,k}^{(\xi^{\prime})})=S_{n,k}\cap(\bigcup\limits_{\xi^{\prime}}T_{n,k}^{(\xi^{\prime})})=S_{n,k}\cap(\bigcup\limits_{\xi^{\prime}}\\{(M_{n,k}^{(\xi^{\prime})}-F_{n,k}^{(\xi^{\prime})})\cup
T_{n,k}^{(\xi^{\prime})}\\})=S_{n,k}\cap(\bigcup\limits_{\xi^{\prime}}f^{-1}(D_{n,k}^{(\xi^{\prime})}))=S_{n,k}\cup
f^{-1}(\bigcup\limits_{\xi^{\prime}}D_{n,k}^{\xi^{\prime}})$ which is
obviously a Baire set because by hypothesis
$\bigcup\limits_{\xi^{\prime}}D_{n,k}^{(\xi^{\prime})}$ is Borel.
Hence for each ($n,k$),
($\bigcup\limits_{\xi}F_{n,k}^{(\xi)})\cup(\bigcup\limits_{\xi}T_{n,k}^{(\xi)})$
is meager. Since by condition (ii), given any open set $H$, there exists
$\Sigma\subseteq\mathbb{N}\times\mathbb{N}$ such that
$H=\bigcup\limits_{(n,k)\in\Sigma}\bigcup\limits_{\xi^{\prime}}D_{n,k}^{(\xi^{\prime})}$,
therefore, upon setting
$P=\bigcup\limits_{(n,k)\in\Sigma}\bigcup\limits_{\xi}(F_{n,k}^{(\xi)}\cup
T_{n,k}^{(\xi)})$ which is meager, we may write
$f_{|X-P}^{-1}(H)=(\bigcup\limits_{(n,k)\in\Sigma}\bigcup\limits_{\xi^{\prime}}M_{n,k}^{\xi^{\prime}})-P$
which may be finally expressed as
$f_{|X-P}^{-1}(H)=\bigcup\limits_{n}(\bigcup\mathcal{C}_{H}^{(n)})-P$ where
$\bigcup\limits_{n}$ is taken over a subset of $\mathbb{N}$ and for each $n$
in that subset, the symbol $\bigcup\mathcal{C}_{H}^{(n)}$ stands for the union
of a subfamily of disjoint regions. ∎

A variant of corollary 2.5 can be obtained in a Marczewski structure. It may
be easily checked (Theorem 33, I, E, Ch 5, [9]) that the pair
($\mathbb{R},\mathcal{P}$) where $\mathbb{R}$ is the real line and
$\mathcal{P}$ denotes the family of all perfect sets in $\mathbb{R}$ is a
perfect base. In this category base, the singular, meager, abundant and Baire
sets constitute the Marczewski classification of sets [9]. The Baire sets for
this category base are called the Marczewski ($s$) sets, the singular, meager
($s_{0}$) and abundant sets are called Marczewski singular, Marczeswki meager
and Marczewski abundant sets respectively.
In the category base ($\mathbb{R},\mathcal{P}$), let
$\mathcal{A}=\\{A_{\alpha}:\alpha\in\Lambda\\}$ be a family of mutually
disjoint $s_{0}$-sets such that their union
$\bigcup\limits_{\alpha\in\Lambda}A_{\alpha}\notin s_{0}$. Let
$B=\bigcup\limits_{\alpha\in\Lambda}A_{\alpha}$ and suppose $B$ is abundant
everywhere in a perfect set $P$. To start with, we show that the cardinality
of the family $\mathcal{A}_{P}=\\{A_{\alpha}\cap
P:A_{\alpha}\in\mathcal{A}\\}$ is $c$ (the cardinality of the continuum).
Divide $\mathcal{A}$ into disjoint subfamilies $\mathcal{A}_{0}$ and
$\mathcal{A}_{1}$ and choose perfect sets $P_{0}$, $P_{1}$ ($\subseteq P$)
such that $B\cap P_{0}$, $B\cap P_{1}$ are abundant, $B\cap
P_{0}\subseteq\bigcup\mathcal{A}_{0}$ and $B\cap
P_{1}\subseteq\bigcup\mathcal{A}_{1}$. Suppose at the n-th stage we have
already constructed subfamilies {$\mathcal{A}_{h}:h\in 2^{n}$} such that $-$

  1. (i)

$\mathcal{A}_{h}\subseteq\mathcal{A}_{h_{|n-1}}$.

  2. (ii)

$\mathcal{A}_{h}\cap\mathcal{A}_{h^{\prime}}=\phi$ if $h,h^{\prime}(\in
2^{n})$, $h\neq h^{\prime}$.

  3. (iii)

there are disjoint perfect sets $P_{h}\subseteq P_{h_{|n-1}}$ such that $B\cap
P_{h}$ is abundant and $B\cap P_{h}\subseteq\bigcup\mathcal{A}_{h}$.

Now fix $h\in 2^{n}$ and divide $\mathcal{A}_{h}$ into disjoint subfamilies
$\mathcal{A}_{h^{\wedge}{0}}$ and $\mathcal{A}_{h^{\wedge}{1}}$ such that for
some disjoint perfect sets $P_{h^{\wedge}{0}},P_{h^{\wedge}{1}}\subseteq
P_{h}$ both $B\cap P_{h^{\wedge}{0}}$ and $B\cap P_{h^{\wedge}{1}}$ are
abundant; $B\cap P_{h^{\wedge}{0}}\subseteq\bigcup\mathcal{A}_{h^{\wedge}{0}}$
and $B\cap P_{h^{\wedge}{1}}\subseteq\bigcup\mathcal{A}_{h^{\wedge}{1}}$.
If we continue this construction, we obtain a collection
{$\mathcal{A}_{f}:f\in 2^{\omega}$} of subfamilies satisfying the following
set of properties :

  1. (i)

$\mathcal{A}_{f_{|n}}\subseteq\mathcal{A}_{f_{|m}}$ for any $f\in 2^{\omega}$
and $m<n$.

  2. (ii)

$\mathcal{A}_{f_{|n}}\cap\mathcal{A}_{f^{\prime}_{|n}}=\phi$ for any
$f,f^{\prime}\in 2^{\omega}$, $f\neq f^{\prime}$.

  3. (iii)

$\bigcup\\{\mathcal{A}_{f_{|n}}:f\in 2^{\omega}\\}=\bigcup\mathcal{A}$.

  4. (iv)

there are disjoint sets $P_{f_{|n}}\subseteq P_{f_{|n-1}}$ such that $B\cap
P_{f_{|n}}$ is abundant and $B\cap
P_{f_{|n}}\subseteq\bigcup\mathcal{A}_{f_{|n}}$.

Let $Q=\\{\bigcap\limits_{n=1}^{\infty}P_{f_{|n}}:f\in 2^{\omega}\\}$.
Then $Q$ is a perfect set contained in $P$ and card($B\cap Q)=c$, for
otherwise since every perfect set is the disjoint union of $c$ number of
perfect sets, there would exist a perfect set $T\subseteq Q$ such that $B\cap
T=\phi$ contradicting the fact that $B$ is abundant everywhere in $P$. Also
our process of construction suggests that the cardinality of the set
{$\alpha\in\Lambda:A_{\alpha}\cap Q\neq\phi$} is $c$.
Let {$P_{\alpha}:\alpha<2^{\omega}$} be an enumeration of all perfect subsets
of $P$ and for each $\alpha\in\Lambda$, let
$\Lambda_{\alpha}=\\{\beta\in\Lambda:A_{\beta}\cap P_{\alpha}\neq\phi$}. From
above, the cardinality of each $\Lambda_{\alpha}$ is $c$. So that we can
choose disjoint sets
$A_{\alpha}^{0},A_{\alpha}^{1}\in\\{A_{\beta}:\beta\in\Lambda_{\alpha}\\}-(\\{A_{\gamma}^{0}:\gamma<\alpha\\}\cup\\{A_{\gamma}^{1}:\gamma<\alpha\\})$
and form disjoint subfamilies {$A_{\alpha}^{0}:\alpha\in\Lambda$} and
{$A_{\alpha}^{1}:\alpha\in\Lambda$}. Let
$C=\bigcup\\{A_{\alpha}^{0}:\alpha\in\Lambda\\}$ and
$D=\\{A_{\alpha}^{1}:\alpha\in\Lambda\\}$. By virtue of the construction, it
can be checked easily that $C$ and $D$ are union of disjoint subfamilies of
{$A_{\alpha}:\alpha\in\Lambda$} which cannot be separated by Baire sets in the
Marczewski base ($\mathbb{R},\mathcal{P}$). Consequently, $C$ and its
complement cannot be separated by Baire sets. Hence, utilizing a Bernstein
type of construction, we may derive that -

######  Theorem 2.7.

In a Marczewski base ($\mathbb{R},\mathcal{C}$), if
{$A_{\alpha}:\alpha\in\Lambda$} is a family of $s_{0}$-sets whose union
$\bigcup\limits_{\alpha\in\Lambda}A_{\alpha}$ is not $s_{0}$, then there exist
disjoint sets $\Lambda_{0}$ and $\Lambda_{1}$ such that
$\Lambda=\Lambda_{0}\cup\Lambda_{1}$ and the disjoint unions
$\bigcup\limits_{\alpha\in\Lambda_{0}}A_{\alpha}$ and
$\bigcup\limits_{\alpha\in\Lambda_{1}}A_{\alpha}$ cannot be separated by
$s$-sets.

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