BABAR-PUB-23/03
SLAC-PUB-17731

# Search for Evidence of Baryogenesis and Dark Matter in
$B^{+}\rightarrow\psi_{\text{D}}+\text{p}$ Decays at BABAR

J. P. Lees  V. Poireau  V. Tisserand  E. Grauges  A. Palano  G. Eigen  D. N.
Brown  Yu. G. Kolomensky  M. Fritsch  H. Koch  R. Cheaib  C. Hearty  T. S.
Mattison  J. A. McKenna  R. Y. So  V. E. Blinov  A. R. Buzykaev  V. P.
Druzhinin  E. A. Kozyrev  E. A. Kravchenko  S. I. Serednyakov  Yu. I. Skovpen
E. P. Solodov  K. Yu. Todyshev  A. J. Lankford  B. Dey  J. W. Gary  O. Long
A. M. Eisner  W. S. Lockman  W. Panduro Vazquez  D. S. Chao  C. H. Cheng  B.
Echenard  K. T. Flood  D. G. Hitlin  Y. Li  D. X. Lin  S. Middleton  T. S.
Miyashita  P. Ongmongkolkul  J. Oyang  F. C. Porter  M. Röhrken  B. T. Meadows
M. D. Sokoloff  J. G. Smith  S. R. Wagner  D. Bernard  M. Verderi  D. Bettoni
C. Bozzi  R. Calabrese  G. Cibinetto  E. Fioravanti  I. Garzia  E. Luppi  V.
Santoro  A. Calcaterra  R. de Sangro  G. Finocchiaro  S. Martellotti  P.
Patteri  I. M. Peruzzi  M. Piccolo  M. Rotondo  A. Zallo  S. Passaggio  C.
Patrignani  B. J. Shuve  H. M. Lacker  B. Bhuyan  U. Mallik  C. Chen  J.
Cochran  S. Prell  A. V. Gritsan  N. Arnaud  M. Davier  F. Le Diberder  A. M.
Lutz  G. Wormser  D. J. Lange  D. M. Wright  J. P. Coleman  D. E. Hutchcroft
D. J. Payne  C. Touramanis  A. J. Bevan  F. Di Lodovico  G. Cowan  Sw.
Banerjee  D. N. Brown  C. L. Davis  A. G. Denig  W. Gradl  K. Griessinger  A.
Hafner  K. R. Schubert  R. J. Barlow  G. D. Lafferty  R. Cenci  A. Jawahery
D. A. Roberts  R. Cowan  S. H. Robertson  R. M. Seddon  N. Neri  F. Palombo
L. Cremaldi  R. Godang  D. J. Summers  G. De Nardo  C. Sciacca  C. P. Jessop
J. M. LoSecco  K. Honscheid  A. Gaz  M. Margoni  G. Simi  F. Simonetto  R.
Stroili  S. Akar  E. Ben-Haim  M. Bomben  G. R. Bonneaud  G. Calderini  J.
Chauveau  G. Marchiori  J. Ocariz  M. Biasini  E. Manoni  A. Rossi  G.
Batignani  S. Bettarini  M. Carpinelli  G. Casarosa  M. Chrzaszcz  F. Forti
M. A. Giorgi  A. Lusiani  B. Oberhof  E. Paoloni  M. Rama  G. Rizzo  J. J.
Walsh  L. Zani  A. J. S. Smith  F. Anulli  R. Faccini  F. Ferrarotto  F.
Ferroni  A. Pilloni  C. Bünger  S. Dittrich  O. Grünberg  T. Leddig  C. Voß
R. Waldi  T. Adye  F. F. Wilson  S. Emery  G. Vasseur  D. Aston  C. Cartaro
M. R. Convery  W. Dunwoodie  M. Ebert  R. C. Field  B. G. Fulsom  M. T. Graham
C. Hast  P. Kim  S. Luitz  D. B. MacFarlane  D. R. Muller  H. Neal  B. N.
Ratcliff  A. Roodman  M. K. Sullivan  J. Va’vra  W. J. Wisniewski  M. V.
Purohit  J. R. Wilson  S. J. Sekula  H. Ahmed  N. Tasneem  M. Bellis  P. R.
Burchat  E. M. T. Puccio  J. A. Ernst  R. Gorodeisky  N. Guttman  D. R. Peimer
A. Soffer  S. M. Spanier  J. L. Ritchie  J. M. Izen  X. C. Lou  F. Bianchi  F.
De Mori  A. Filippi  L. Lanceri  L. Vitale  F. Martinez-Vidal  A. Oyanguren
J. Albert  A. Beaulieu  F. U. Bernlochner  G. J. King  R. Kowalewski  T. Lueck
C. Miller  I. M. Nugent  J. M. Roney  R. J. Sobie  T. J. Gershon  P. F.
Harrison  T. E. Latham  S. L. Wu

###### Abstract

A new dark sector anti-baryon, denoted $\psi_{\text{D}}$, could be produced in
decays of $B$ mesons. This letter presents a search for
$B^{+}\rightarrow\psi_{\text{D}}+\text{p}$ (and the charge conjugate) decays
in $e^{+}e^{-}$ annihilations at 10.58 GeV, using data collected in the BABAR
experiment. Data corresponding to an integrated luminosity of 398 fb-1 are
analyzed. No evidence for a signal is observed. Branching fraction upper
limits in the range from $10^{-7}$ – $10^{-5}$ are obtained at 90$\%$
confidence level for masses of 1.0 $<m_{\psi_{\text{D}}}<$ 4.3 GeV/c2. The
result is also reinterpreted to provide the first limits on a supersymmetric
model with R-parity violation and a light neutralino.

dark matter

††thanks: Deceased

The BABAR Collaboration

The existence of dark matter (DM) is established from astrophysical
observations [1, 2, 3]. Measurements of the cosmic microwave background (CMB)
by the Planck satellite [4] have shown that only $\sim$15$\%$ of the matter
content of the universe can be accounted for from Standard Model (SM)
particles. The remaining fraction is referred to as DM. Understanding the mass
scale and nature of DM is one of the most pressing issues of modern particle
physics.

Another pressing issue is understanding the baryon asymmetry of the universe
(BAU) [5]. A dynamical mechanism, baryogenesis, is required to produce an
initial excess of baryons over anti-baryons consistent with CMB and big-bang
nucleosynthesis (BBN) measurements [6, 7].

In Ref. [8] a new dark sector anti-baryon111The charge conjugate involves a
dark sector baryon accompanied by an anti-proton, both channels are used in
this analysis. This is implied throughout., $\psi_{\text{D}}$, is proposed,
which can also explain the BAU. In this model, baryogenesis occurs due to out-
of-thermal-equilibrium production of $b$ and $\bar{b}$ quarks in the early
universe through the decay of a massive, long-lived scalar field. The $b$ and
$\bar{b}$ quarks hadronize into $B^{0}_{s},B^{0}$, and $B^{\pm}$ mesons. The
$B^{0}$ – $\bar{B}^{0}$ mesons then undergo CP-violating oscillations before
decaying into a SM baryon $\mathcal{B}$, $\psi_{\text{D}}$, and any number of
additional light mesons $\mathcal{M}$. These CP-violating oscillations can
originate from the SM or beyond the standard model (BSM) processes. The term
$B$-mesogenesis is coined to describe this mechanism. Decays of $B$ mesons
into $\psi_{\text{D}}$ are mediated by new particles introduced at the TeV
scale. In this scenario, matter-antimatter asymmetries are generated in the
visible and dark sectors with equal magnitudes but opposite signs, keeping the
total baryon number conserved. Current bounds on the semi-leptonic charge
asymmetry in the decays of $B^{0}_{s}$ and $B^{0}$ set a lower bound on the
total branching fraction $BF(B\rightarrow{\cal B}\psi_{\text{D}}{\cal
M})\gtrsim 10^{-4}$ assuming that the observed baryon-antibaryon-asymmetry is
explained solely by the mesogenesis mechanism of Ref. [9].

We present herein a search for the exclusive decay
$B^{+}\rightarrow\psi_{\text{D}}+\text{p}$ and its charge conjugate. We
utilize the hadronic recoil $B$-tagging method as outlined in Ref. [10]. One
of the $B$ mesons from $e^{+}e^{-}\rightarrow B^{+}B^{-}$ is fully
reconstructed from known hadronic decay modes, and is referred to as the
$B_{\text{tag}}$ [10]. The rest of the event222All other tracks and clusters
in the event., which must include the proton, is then assigned to the other
$B$ meson, denoted as the $B_{\text{sig}}$. Previous limits have been provided
from a reinterpretation of a search for decays of $b$-flavored hadrons with
large missing energy at LEP [9, 11]. In addition, direct searches for the TeV-
scale mediator at the LHC [12, 13], and DM stability, require
$0.94<m_{\psi_{\text{D}}}<3.5$ GeV/c2 [9].

Constraints on exclusive decays (with a single SM baryon in the final state)
are calculated using phase-space considerations for different baryons [9]. The
results depend on the effective operators
$\mathcal{O}_{i,j}=(\psi_{\text{D}}b)(q_{i}q_{j})$ mediating the decay, where
$i$ and $j$ specify the quark content, $q_{i}=u,c$ and $q_{j}=d,s$. There are
four possible flavor-combination operators of interest for $B$ meson decays.
The decay presented here probes $\mathcal{O}_{ud}$. New limits on
$B^{0}\rightarrow\psi_{\text{D}}+\Lambda$ from BABAR are presented in Ref.
[14], which probes the $\mathcal{O}_{us}$ operator. Since the presented search
is not sensitive to the Dirac or Majorana nature of the invisible particle, it
is potentially sensitive to other models predicting
$B^{+}\rightarrow\text{invisible}+\text{p}$ . In our conclusion, we also
reinterpret the search for a supersymmetric model with R-parity violation and
a light neutralino [15]. In addition, since we seek a charged final state, the
result could also be reinterpreted as a search specifically for charged
$B$-mesogenesis, as described in Ref. [16]. Charged $B$-mesogenesis scenarios
are being actively probed at several collider-based and neutrino experiments.
The result from the present work can provide a relevant constraint for these
studies.

The BABAR detector is described in Refs. [17, 18] and consists of several
subsystems arranged in a cylindrical structure around the $e^{+}e^{-}$
interaction point. Charged-particle momenta are measured by a five-layer
double-sided silicon vertex tracker and a 40-layer multi-wire drift chamber,
both operating in the 1.5 T magnetic field of a superconducting solenoid. The
particle identification (PID) for protons, kaons, and pions uses the specific
energy loss measured in the tracking detectors and the measurement of the
Cherenkov angle provided by the internally reflecting, ring-imaging Cherenkov
detector. Photons are detected in the electromagnetic calorimeter (EMC). Muon
identification is provided by the instrumented flux return. Protons are
identified using BABAR likelihood-based particle identification algorithms
detailed in Ref. [19]. There is a negligible difference in the reconstruction
efficiencies of protons and anti-protons.

The data sample used corresponds to an integrated luminosity of 398.5 fb-1
[19] collected at the PEP-II $e^{+}e^{-}$ storage ring at SLAC. A further 32.5
fb-1 is used to optimize the analysis strategy and is excluded from the sample
used to obtain the final result. At PEP-II, 9 GeV electrons collide with 3.1
GeV positrons at center-of-mass (CM) energies near 10.58 GeV ($\Upsilon(4S)$
resonance). The average cross section for $B^{+}B^{-}$ pair production of
electron-positron annihilation is $\sigma(e^{+}e^{-}\rightarrow
B^{+}B^{-})\sim 550$ pb; thus the data sample corresponds to $\sim 2\times
10^{8}$ produced $B^{+}B^{-}$ pairs.

Monte Carlo (MC) generators are used to simulate background events that
emanate from inclusive $e^{+}e^{-}\rightarrow B\bar{B}$ (EVTGEN [20]) or
continuum $e^{+}e^{-}\rightarrow q\bar{q}\>\>(q=udsc)$ processes (JETSET [21,
22]). Signal events are generated using EVTGEN. Samples were made for eight
different $\psi_{\text{D}}$ mass hypotheses: 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0
and 4.2 GeV/$c^{2}$. The propagation of particles through the detector is
simulated using the GEANT4 toolkit [23].

The reconstructed $B_{\text{tag}}$ must have a CM energy
($E^{*}_{B_{\text{tag}}}$) within $\pm$0.2 GeV of the beam energy,
$E^{*}_{\text{beam}}$, in the CM frame. The energy-substituted mass is defined
as
$m_{\text{ES}}c^{2}=\sqrt{E_{\text{beam}}^{\>*2}-\vec{\text{p}}^{\>*2}_{B_{\text{tag}}}c^{2}}$,
where $\vec{\text{p}}^{\>*}_{B_{\text{tag}}}$ is the three-momentum of
$B_{\text{tag}}$ in the CM frame. We require $m_{\text{ES}}$ of the
$B_{\text{tag}}$ to lie within the nominal $B^{+}$ mass range defined by
$5.27-5.29$ GeV/c2. When multiple $B_{\text{tag}}$ candidates are found in one
event, the one that has the lowest value of $\Delta
E=E^{*}_{\text{beam}}-E^{*}_{B_{\text{tag}}}$ is selected.

On the signal side, the presence of one and only one charged track is
required, and it must be consistent with the proton hypothesis. To suppress
the remaining inclusive background, we use a single multivariate classifier
based on a boosted decision tree (BDT) algorithm which is trained on the
combined background and signal MC samples. The BDT includes the following
kinematic variables from the $B_{\text{tag}}$: $\Delta E$ and
${m_{\text{ES}}}$; information about the hadronic decay channel and its purity
[10]; and the magnitude of the thrust vector, defined as the sum of the
magnitudes of the momenta of all tracks and calorimeter clusters projected
onto the thrust axis [10]. The following features from the $B_{\text{sig}}$
are also included: the total extra neutral energy on the signal side in the CM
frame; the cosine of the polar angle of the missing momentum vector recoiling
against the $B_{\text{tag}}$ meson and the signal candidate in the laboratory
frame; the number of neutral particles and the number of $\pi^{0}$ candidates
on the signal side, where a $\pi^{0}$ candidate is two photons with an
invariant mass within $15$ MeV/c2 of the nominal $\pi^{0}$ mass (134.9 MeV/c2
[6]). Additional features include the ratio of the second to zeroth Fox-
Wolfram [24] moment for all tracks and neutral clusters (denoted as $R_{2}$),
and the cosine of the thrust vector. These features are uncorrelated (in most
cases $\ll 50\%$) for both signal and background events. The features that
provide the most discriminating power are $m_{ES}$, purity and decay
information, $R_{2}$ and thrust vector magnitude. An additional criterion that
no neutral pion candidates should present on the signal side is applied before
the final analysis, at which point no extra neutral candidates remain.

Figure 1 shows the distribution of the BDT responses ($\nu_{\text{BDT}}$).
Events are required to have $\nu_{\text{BDT}}>0.95$, which retains $>$ 99$\%$
of all the simulated signals and 0.0028$\%$ of the simulated background.

Figure 1:  BDT response for data and all backgrounds. The signal shown is an
inclusive signal sample including all 8 simulated signal samples.

Figure 2 shows the distribution of $m_{\text{ES}}$ for inclusive MC
background, signal, and data. The signal events peak around the nominal $B$
meson mass and background events are dominated by the continuum events.

Known discrepancies in the simulation [25] of the $B\bar{B}$ and of $q\bar{q}$
events are corrected for, in a two-stage process, based on an analysis of the
distribution of $R_{2}$ (Fig. 3). First, a correction factor for the
$q\bar{q}$ samples, $f_{q\bar{q}}=1.05\pm 0.03$, is extracted from the
$R_{2}>0.7$ region. Similarly, a correction factor, $f_{B\bar{B}}=0.85\pm
0.07$ for the $B\bar{B}$ samples is extracted from the $R_{2}<0.7$ region,
assuming an equal contribution to the correction from both $B^{0}\bar{B}^{0}$
and $B^{+}\bar{B}^{-}$. In both cases, the uncertainties are purely
statistical. Under the assumption that $f_{B\bar{B}}$ is independent of the
$B_{\text{sig}}$ decay mode, the signal efficiency is also re-scaled by
$f_{B\bar{B}}$.

Figure 2:  The energy-substituted mass ($m_{\text{ES}}$) of the
$B_{\text{tag}}$ candidate for MC background processes and data. An example
signal distribution is shown with arbitrary normalization (no correction
applied). Figure 3:  Simulated distributions of the ratio of the second-to-
zeroth Fox-Wolfram moment for all tracks (denoted as $R_{2}$).

The signal efficiencies are extracted as the ratios of selected events to the
total generated from the eight simulated signal samples. The signal efficiency
varies from 0.00145 for $m_{\psi_{\text{D}}}=1.0$ GeV/c2 to 0.0006 for
$m_{\psi_{\text{D}}}=4.2$ GeV/c2. The largest loss of efficiency comes from
use of the standard BABAR reconstruction algorithm and the requirements of a
proton track, with no accompanying charged particles on the signal side. The
efficiencies extracted from the eight signal samples are fitted with a smooth
seventh-order polynomial, with $\chi^{2}/ndf=0.98$, to allow interpolation at
any intermediate mass hypothesis.

The missing-mass ($m_{\text{miss}}$), which in the case of a signal would be
the $\psi_{\text{D}}$ mass, is calculated from the four momenta of the signal
$B_{\text{sig}}$ and proton:

$m_{\text{\text{miss}}}c^{2}=\sqrt{(E^{\>*}_{B_{\text{sig}}}-E^{\>*}_{\text{p}})^{2}-|\vec{\text{p}}^{\>*}_{B_{\text{sig}}}-\vec{\text{p}}^{\>*}_{\text{p}}|^{2}c^{2}}$
(1)

where ($\vec{\text{p}}^{\>*}_{B_{\text{sig}}},E^{\>*}_{B_{\text{sig}}}$) and
($\vec{\text{p}}^{\>*}_{\text{p}},E^{\>*}_{\text{p}}$) are the four-momenta of
the signal $B_{\text{sig}}$ and proton, respectively, in the CM frame. Figure
4 shows the missing-mass distribution for the data, background, and an example
signal hypothesis after all selection criteria have been applied. For each
signal mass, the missing-mass distribution is fitted with a double-sided
Crystal Ball [26, 27] function to extract the signal mass resolution. The
resolution is obtained from the fits to the signal MC and defined as
$\sigma_{m}$ = FWHM/2.35; it varies from $\sim$110 MeV/c2 at
$m_{\text{\text{miss}}}$ = 1 GeV/c2 to $\sim$11 MeV/c2 at $m_{\text{miss}}$ =
4.2 GeV/c2. The resolutions for all mass values in the search region are
interpolated from the fit to the eight signal samples using an exponential
function, the $\chi^{2}/ndf$ of the fit was 1.1.

A scan is performed across the missing-mass distribution with a step size
equal to the signal mass resolution ($\sigma_{m}$) interpolated from fits to
the signal MC samples. In total 127 mass hypotheses were considered in the
range $1.0<m_{\text{miss}}<4.29$ GeV/c2.

Figure 4:  Missing-mass distributions after all selections are applied for a
simulated signal sample with $m_{\psi_{\text{D}}}=$ 2 GeV/c2 (solid green
line), inclusive SM background (stack histograms) and data (black dots). In
total 46 events remain in the data and 48 remain in the inclusive SM MC
sample.

The largest systematic uncertainty comes from the data/MC correction (8.2$\%$)
and affects the signal efficiency. The uncertainty on the correction factor
includes several contributions including imperfections in the modeling of
reconstruction and particle identification. In addition, there are
normalization uncertainties in the yield of $B^{+}B^{-}$ pairs which include:
the uncertainty on the number of $\Upsilon(4S)$ mesons (0.6$\%$[28]); the
uncertainty on the $\Upsilon(4S)\rightarrow B^{+}B^{-}$ branching fraction
(1.2$\%$); and, the uncertainty on the signal efficiency due to the PID
algorithms incorrectly identifying a proton/anti-proton track (1$\%$). The
total uncertainty on the signal efficiency is 8.4 $\%$.

In the absence of a signal, 90$\%$ confidence level (C.L.) upper limits on the
branching fractions are derived using a profile likelihood method [29]. A
Poisson counting approach is followed using only the data. The number of
signal and background events are assumed to follow Poisson distributions, and
the efficiency is assumed Gaussian with a standard deviation equal to the
total systematic uncertainty. For a given $\psi_{\text{D}}$ mass hypothesis,
the signal region is defined in the data as the region
$m_{\psi_{\text{D}}}-5\sigma_{m}<m_{\text{miss}}<m_{\psi_{\text{D}}}+5\sigma_{m}$,
the side-bands ($[+5\sigma,+10\sigma]$ and $[-10\sigma,-5\sigma]$) on either
side of this window are classified as the background region.

Figure 5 shows the resulting 90 $\%$ C.L. upper limit on the branching
fraction. The largest local significance is 3.5 $\sigma$ at 3.3 GeV/c2 which
results in a 1 $\sigma$ global significance. Almost all the available
parameter space for the $\mathcal{O}^{2,3}_{ud}$ operators is constrained with
the BABAR data set. However, operator $\mathcal{O}^{1}_{ud}$ remains mostly
unconstrained between 1.9 – 3.0 GeV/c2 and below 1.5 GeV/c2.

Figure 5: Derived 90 $\%$ C.L. upper limits on the branching fraction
$B^{+}\rightarrow\psi_{\text{D}}+\text{p}$ and the charge conjugate for BABAR
data set corresponding to 398 fb-1. The theory expectation for the three
effective operators are from Ref. [9].

Our result can be reinterpreted to constrain other models with missing mass in
the final state, including the RPV supersymmetry process
$B^{+}\rightarrow\tilde{\chi}_{0}+p$, where $\tilde{\chi}_{0}$ is the lightest
neutralino. In Fig. 6 the branching fraction upper limits obtained in the
present analysis are converted to limits on the RPV coupling
$\lambda_{113}^{{}^{\prime\prime}}$ divided by the relevant squark mass
squared as a function of the neutralino mass. These are unique limits, there
are no previous results for this channel.

Figure 6: Derived 90 $\%$ C.L. upper limits for BABAR data set corresponding
to 398 fb-1 on the RPV coupling $\lambda_{113}^{{}^{\prime\prime}}$ for the
process $B^{+}\rightarrow\tilde{\chi}^{0}+p$ using the conversion factors
presented in Fig. 2 of Ref. [15].

To summarize, a search for $B^{+}\rightarrow\psi_{\text{D}}+\text{p}$ has been
presented. This is the first attempt to directly search for this channel. No
signal is observed, and 90$\%$ C.L upper limits from $10^{-7}$ – $10^{-5}$ are
set on the branching fraction. A large fraction of the $B$-mesogenesis
parameter space is excluded by this measurement. Our result also constrains
the branching fraction upper limits on the RPV coupling,
$\lambda_{113}^{{}^{\prime\prime}}$, divided by the relevant squark mass
squared as a function of the neutralino mass, at the level $10^{-7}$ –
$10^{-6}$ for 0.5$<m_{\tilde{\chi}^{0}}<$ 4.29 GeV/c2. In addition, we note
that the limits (outlined in the Supplementary Material) can also be
reinterpreted to provide constraints on other models e.g. charged
$B$-mesogenesis.

We are grateful for the extraordinary contributions of our PEP-II colleagues
in achieving the excellent luminosity and machine conditions that have made
this work possible. The success of this project also relies critically on the
expertise and dedication of the computing organizations that support BABAR,
including GridKa, UVic HEP-RC, CC-IN2P3, and CERN. The collaborating
institutions wish to thank SLAC for its support and the kind hospitality
extended to them. We also wish to acknowledge the important contributions of
J. Dorfan and our deceased colleagues E. Gabathuler, W. Innes, D.W.G.S. Leith,
A. Onuchin, G. Piredda, and R. F. Schwitters.

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# Primal Topological Spaces

Santanu Acharjee1, Murad Özkoç2, Faical Yacine Issaka3
1Department of Mathematics
Gauhati University
Guwahati-781014, Assam, India
2Muğla Sıtkı Koçman University
Faculty of Science
Department of Mathematics
48000, Menteşe-Muğla, Turkey
3Muğla Sıtkı Koçman University
Graduate School of Applied and Natural Sciences
Mathematics, 48000, Menteşe-Muğla, Turkey
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
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###### Abstract

The purpose of this paper is to introduce a new structure ‘primal’. Primal is
dual to grill. Like ideal, dual of filter, this new structure also generates a
new topology named ‘primal topology’. We introduce a new operator using
primal, which satisfies Kuratowski’s closure axioms. Mainly, we prove that
primal topology is finer than the topology of a primal topological space. We
provide structure of base of primal topology and prove other fundamental
results related to this new structure.

2020 AMS Classifications: 54A05; 54B99; 94A60.

Keywords: Primal, grill, primal topological space, topological cryptography,
Kuratowski’s closure, base.

##  1 Introduction

Topology is one of the branches of mathematics which is highly applicable
[19]. Due to its applicability in both science and social science, several new
ideas have been developed in topology with classical structures. Kuratowski
introduced the idea of ideal from filter [20]. One may consider ideal as the
dual of filter. Similarly, one of the classical structures of topology is
grill. The definition of grill was introduced by Choquet [13] in the year
1947.

Later, Thron [3] introduced proximity structures in grills. In 1977,
Chattopadhyay and Thron [1] extended ideas of closure space with grills.
Moreover, Chattopadhyay et al. [2] extended ideas of grills to study merotopic
spaces. Since then, the structure of grill has been highly used in topology.
Roy and Mukherjee [4, 5, 6] studied various topological properties with
grills. Operators based on grills were introduced by Roy et al. [7], Nasef and
Azzam [11], and many others. Modak [8, 9] studied grill-filter space and
related properties. Hosny [10] studied grill structures in $\delta$-set.
Cluster systems via grills were studied in [12]. Moreover, various advanced
results with grills were studied in [14, 15, 16, 17, 18] and many others. But,
it is important to note that the literature on grill structures are less
comparison to filter, ideal, etc. Moreover, interdisciplinary applications of
grills are rare to be found. Janković and Hamlett [21] introduced a new
topological space using ideal from a given topology of a topological space.
Since, primal is also dual structure of grill, thus we are motivated by
Janković and Hamlett [21] to introduce a new topology using primal.

In this paper, we introduce the dual structure of grill named ‘primal’.
Moreover, we introduce a new topology named “primal topology” and study
several fundamental properties. In 1990, Isham [24] connected general topology
with quantum topology. In this paper, we also observe scopes of some quantum
behavior of a new structure $(.)^{\diamond}$ introduced in Definition 3.3.
Similarly, uncertain behaviors can be observed in Theorem 3.10 and results
next to it. As of our knowledge is concerned, this paper is the first paper in
the literature of topology to introduce dual structure of grill, thus we are
confined to study fundamental results only related to primal.

##  2 Preliminaries

Throughout this present paper, $(X,\tau)$ and $(Y,\sigma)$ (briefly, $X$ and
$Y$) represent topological spaces unless otherwise stated. For any subset $A$
of a space $X$, $cl(A)$ and $int(A)$ denote closure and t interior of $A$,
respectively. The powerset of a set $X$ will be denoted by $2^{X}.$ The family
of all open neighborhoods of a point $x$ of $X$ is denoted by $\tau(x).$ Now,
we procure the following definition of grill.

######  Definition 2.1.

[13] A family $\mathcal{G}$ of $2^{X}$ is called a grill on $X$ if
$\mathcal{G}$ satisfies the following conditions:

$(1)$ $\emptyset\notin\mathcal{G},$

$(2)$ if $A\in\mathcal{G}$ and $A\subseteq B,$ then $B\in\mathcal{G}$,

$(3)$ if $A\cup B\in\mathcal{G},$ then $A\in\mathcal{G}$ or $B\in\mathcal{G}.$

Since, primal is dual to grill, thus we skip to add other results on grill,
otherwise it will unnecessarily increase the volume of this paper.

##  3 Primal and a new topological structure

In this section, we introduce a new structure in topology. This new structure
is called primal. Primal is a dual structure of grill. Now, we have the
following definition of primal.

######  Definition 3.1.

Let $X$ be a nonempty set. A collection $\mathcal{P}\subseteq 2^{X}$ is called
a primal on $X$ if it satisfies the following conditions:

(i) $X\notin\mathcal{P}$,

(ii) if $A\in\mathcal{P}$ and $B\subseteq A$, then $B\in\mathcal{P}$,

(iii) if $A\cap B\in\mathcal{P}$, then $A\in\mathcal{P}$ or $B\in\mathcal{P}$.

######  Corollary 3.0.1.

Let $X$ be a nonempty set. A collection $\mathcal{P}\subseteq 2^{X}$ is a
primal on $X$ if and only if it satisfies the following conditions:

(i) $X\notin\mathcal{P}$,

(ii) if $B\notin\mathcal{P}$ and $B\subseteq A$, then $A\notin\mathcal{P}$,

(iii) if $A\notin\mathcal{P}$ and $B\notin\mathcal{P},$ then $A\cap
B\notin\mathcal{P}.$

Now, let us consider the following examples.

######  Example 3.1.

Let $X=\\{a,b\\}.$ Then, all primals defined on $X$ are
$\mathcal{P}_{1}=\emptyset,$ $\mathcal{P}_{2}=\\{\emptyset,\\{a\\}\\},$
$\mathcal{P}_{3}=\\{\emptyset,\\{b\\}\\}$ and
$\mathcal{P}_{4}=\\{\emptyset,\\{a\\},\\{b\\}\\}.$

######  Example 3.2.

Let $X$ be a nonempty set. It is obvious that the family
$\mathcal{P}=2^{X}\setminus\\{X\\}$ is a primal on $X,$ where $2^{X}$ denotes
the powerset of $X.$

######  Corollary 3.0.2.

It is not difficult to see that $2^{n}$ primals can be written on a set with
$n$ elements.

######  Theorem 3.1.

Let $\mathcal{G}$ be a grill on $X$. Then, $\\{A|A^{c}\in\mathcal{G}\\}$ is a
primal on $X$.

###### Proof.

Let $\mathcal{G}$ be a grill of $X$ and
$\mathcal{P}=\\{A|A^{c}\in\mathcal{G}\\}$. Then, we are to show that
$\mathcal{P}$ is a primal.

$(i)$ Since $\emptyset\notin\mathcal{G}$, thus $X\notin\mathcal{P}$.

$(ii)$ Let $A\in\mathcal{P}$ and $B\subseteq A$. Then, $A^{c}\subseteq B^{c}$.
Since $A^{c}\in\mathcal{G}$, thus $B^{c}\in\mathcal{G}$. Hence,
$B\in\mathcal{P}$.

$(iii)$ Let $A\cap B\in\mathcal{P}$. Then, $A^{c}\cup B^{c}=(A\cap
B)^{c}\in\mathcal{G}$. Therefore, we get $A^{c}\in\mathcal{G}$ or
$B^{c}\in\mathcal{G}.$ Hence, $A\in\mathcal{P}$ or $B\in\mathcal{P}$. Thus,
$\mathcal{P}$ is a primal on $X.$ ∎

######  Theorem 3.2.

Let $\mathcal{P}$ and $\mathcal{Q}$ be two primals on $X$. Then,
$\mathcal{P\cup Q}$ is a primal on $X$.

###### Proof.

$(i)$ Since $\mathcal{P}$ and $\mathcal{Q}$ be two primals on $X$, thus we get
$X\notin\mathcal{P}$ and $X\notin\mathcal{Q}$. Hence, $X\notin\mathcal{P\cup
Q}$.

$(ii)$ Let $A\in\mathcal{P\cup Q}$ and $B\subseteq A$. Then, $A\in\mathcal{P}$
or $A\in\mathcal{Q}$. Then, $B\in\mathcal{P}$ or $B\in\mathcal{Q}$. It yields
$B\in\mathcal{P\cup Q}$.

$(iii)$ Let $A\cap B\in\mathcal{P\cup Q}$. Then, $A\cap B\in\mathcal{P}$ or
$A\cap B\in\mathcal{Q}$. If $A\cap B\in\mathcal{P}$, then either
$A\in\mathcal{P}$ or $B\in\mathcal{P}$. Again, if $A\cap B\in\mathcal{Q}$,
then either $B\in\mathcal{Q}$ or $B\in\mathcal{Q}$. Then, obviously
$A\in\mathcal{P\cup Q}$ or $B\in\mathcal{P\cup Q}$.

Thus, $\mathcal{P\cup Q}$ is a primal on $X$. ∎

######  Remark 3.1.

The intersection of two primals defined on $X$ need not be a primal on $X$ as
shown by the following example.

######  Example 3.3.

Let $X=\\{a,b\\}$ with primals $\mathcal{P}=\\{\emptyset,\\{a\\}\\}$ and
$\mathcal{Q}=\\{\emptyset,\\{b\\}\\}$ on $X$. Then, $\mathcal{P\cap
Q}=\\{\emptyset\\}$ is not a primal on $X$ since
$\\{a\\}\cap\\{b\\}=\emptyset\in\mathcal{P\cap Q}$, but neither
$\\{a\\}\in\mathcal{P\cap Q}$ nor $\\{b\\}\in\mathcal{P\cap Q}.$

######  Remark 3.2.

The family of sets formed by the intersection (union) of the elements of two
primals need not be a primal on $X$ as shown by the following example.

######  Example 3.4.

Let $X=\\{a,b\\}$ with primals $\mathcal{P}=\\{\emptyset,\\{a\\}\\}$ and
$\mathcal{Q}=\\{\emptyset,\\{b\\}\\}$ on $X.$

$(a)$ The family $\mathcal{R}=\\{P\cap
Q|(P\in\mathcal{P})(Q\in\mathcal{Q})\\}=\\{\emptyset\\}$ is not a primal on
$X$ since $\\{a\\}\cap\\{b\\}=\emptyset\in\mathcal{R}$ but neither
$\\{a\\}\in\mathcal{R}$ nor $\\{b\\}\in\mathcal{R}.$

$(b)$ The family $\mathcal{S}=\\{P\cup
Q|(P\in\mathcal{P})(Q\in\mathcal{Q})\\}=2^{X}$ is not a primal on $X$, since
$X\in\mathcal{S}.$

######  Definition 3.2.

A topological space $(X,\tau)$ with a primal $\mathcal{P}$ on $X$ is called a
primal topological space and denoted by $(X,\tau,\mathcal{P}).$

######  Example 3.5.

Let $(X,\tau)$ be a topological space, where $X=\\{a,b\\}$,
$\tau=\\{\emptyset,\\{a\\},X\\}$ and $\mathcal{P}=\\{\emptyset,\\{a\\}\\}$
primal on $X.$ Then, $(X,\tau,\mathcal{P})$ is a primal topological space.

Now, we define a new kind of topological operator based on primal. The new
structure is given below.

######  Definition 3.3.

Let $(X,\tau,\mathcal{P})$ be a primal topological space. We consider a map
$(\cdot)^{\diamond}:2^{X}\rightarrow 2^{X}$ as
$A^{\diamond}(X,\tau,\mathcal{P})=\\{x\in X:(\forall U\in\tau(x))(A^{c}\cup
U^{c}\in\mathcal{P})\\}$ for any subset $A$ of $X.$ We can also write
$A_{\mathcal{P}}^{\diamond}$ as $A^{\diamond}(X,\tau,\mathcal{P})$ to specify
the primal as per our requirements.

######  Remark 3.3.

Let $(X,\tau,\mathcal{P})$ be a primal topological space. For any subset $A$
of $X,$ $A^{\diamond}\subseteq A$ or $A\subseteq A^{\diamond}$ need not be
true as shown by the following examples.

######  Example 3.6.

Let $X=\\{a,b,c\\}$ and $\tau=\\{\emptyset,\\{a\\},\\{b\\},\\{a,b\\},X\\}$. We
consider a primal $\mathcal{P}=\\{\emptyset,\\{a\\},\\{b\\},\\{a,b\\}\\}$ on
$X$. Now, if $A=\\{a,b\\},$ then $\emptyset=A^{\diamond}\subseteq
A=\\{a,b\\}.$

######  Example 3.7.

Let $X=\\{a,b,c\\}$ with the discrete topology. We consider a primal
$\mathcal{P}=\\{\emptyset,\\{a\\},\\{b\\},\\{a,b\\}\\}$ on $X.$ Now, if
$A=\\{c\\},$ then $\\{c\\}=A\subseteq A^{\diamond}=X.$

######  Theorem 3.3.

Let $(X,\tau,\mathcal{P})$ be a primal topological space. Then, the following
statements hold for any two subsets of $A$ and $B$ of $X.$

$(i)$ If $A^{c}\in\tau,$ then $A^{\diamond}\subseteq A,$

$(ii)$ $\emptyset^{\diamond}=\emptyset$,

$(iii)$ $cl(A^{\diamond})=A^{\diamond},$

$(iv)$ $(A^{\diamond})^{\diamond}\subseteq A^{\diamond},$

$(v)$ If $A\subseteq B$, then $A^{\diamond}\subseteq B^{\diamond}$,

$(vi)$ $A^{\diamond}\cup B^{\diamond}=(A\cup B)^{\diamond},$

$(vii)$ $(A\cap B)^{\diamond}\subseteq A^{\diamond}\cap B^{\diamond}.$

###### Proof.

$(i)$ Let $A^{c}\in\tau$ and $x\in A^{\diamond}.$ Suppose that $x\notin A.$
Then, $A^{c}\in\tau(x).$ Since $x\in A^{\diamond},$ $A^{c}\cup
U^{c}\in\mathcal{P}$ for all $U\in\tau(x).$ Therefore, $X=A\cup
A^{c}=(A^{c})^{c}\cup A^{c}\in\mathcal{P}.$ This contradicts with
$X\notin\mathcal{P}.$ Hence, $A^{\diamond}\subseteq A.$

$(ii)$ It obvious from (i) since $\emptyset^{c}\in\tau.$

$(iii)$ We have always $A^{\diamond}\subseteq cl(A^{\diamond}).$ Conversely,
let $x\in cl(A^{\diamond})$ and $U\in\tau(x).$ Then, $U\cap
A^{\diamond}\neq\emptyset.$ Therefore, there exists $y\in X$ such that $y\in
U$ and $y\in A^{\diamond}.$ Then, we have $V^{c}\cup A^{c}\in\mathcal{P}$ for
all $V\in\tau(y).$ Thus, we get $U^{c}\cup A^{c}\in\mathcal{P}.$ This means
that $x\in A^{\diamond}.$ Hence, $cl(A^{\diamond})\subseteq A^{\diamond}.$
Therefore, $A^{\diamond}$ is closed in $X.$

$(iv)$ It is obvious from (i) and (iii).

$(v)$ Let $A\subseteq B$ and $x\in A^{\diamond}.$ Then, we have $A^{c}\cup
U^{c}\in\mathcal{P}$ for all $U\in\tau(x).$ Thus, $B^{c}\cup
U^{c}\in\mathcal{P}$ since $A\subseteq B.$ Hence, $x\in A^{\diamond}$. Hence,
$A^{\diamond}\subseteq B^{\diamond}.$

$(vi)$ We can get from (v) that $A^{\diamond}\subseteq(A\cup B)^{\diamond}$
and $B^{\diamond}\subseteq(A\cup B)^{\diamond}$. Hence, $A^{\diamond}\cup
B^{\diamond}\subseteq(A\cup B)^{\diamond}.$ Conversely, let $x\notin
A^{\diamond}\cup B^{\diamond}.$ Then $x\notin A^{\diamond}$ and $x\notin
B^{\diamond}.$ Then, there exist open sets $U$ and $V$ containing $x$ such
that $A^{c}\cup U^{c}\notin\mathcal{P}$ and $B^{c}\cup
V^{c}\notin\mathcal{P}.$ Put $W=U\cap V.$ Hence, $W$ is open containing $x$
such that $A^{c}\cup W^{c}\notin\mathcal{P}$ and $B^{c}\cup
W^{c}\notin\mathcal{P}.$ Then, we have $(A\cup B)^{c}\cup W^{c}=(A^{c}\cap
B^{c})\cup W^{c}=(A^{c}\cup W^{c})\cap(B^{c}\cup W^{c})\notin\mathcal{P}$
since $\mathcal{P}$ is primal. This means that $x\notin(A\cup B)^{\diamond}.$
Thus, $(A\cup B)^{\diamond}\subseteq A^{\diamond}\cup B^{\diamond}.$

$(vii)$ and $(viii)$ can be proven similarly. ∎

######  Remark 3.4.

The inclusion given in (vii) of Theorem 3.3 need not be reversible as shown by
the following example.

######  Example 3.8.

Let $(X,\tau)$ be indiscrete topological space, where $X=\\{a,b,c\\}.$ We
consider a primal
$\mathcal{P}=\\{\emptyset,\\{a\\},\\{b\\},\\{c\\},\\{a,b\\},\\{a,c\\}\\}$ on
$X$. Now, if $A=\\{b\\}$ and $B=\\{c\\},$ then we have $A^{\diamond}\cap
B^{\diamond}=X\cap X=X\neq\emptyset=\emptyset^{\diamond}=(A\cap
B)^{\diamond}.$

######  Theorem 3.4.

Let $(X,\tau,\mathcal{P})$ be a primal topological space and $A,B\subseteq X.$
If $A$ is open in $X,$ then $A\cap B^{\diamond}\subseteq(A\cap B)^{\diamond}.$

###### Proof.

Let $A\in\tau$ and $x\in A\cap B^{\diamond}.$ Therefore, $x\in A$ and $x\in
B^{\diamond}.$ Then, we have $B^{c}\cup U^{c}\in\mathcal{P}$ for all
$U\in\tau(x).$ Since $A\in\tau,$ we get $(A\cap B)^{c}\cup
U^{c}=B^{c}\cup(A\cap U)^{c}\in\mathcal{P}$ for all $U\in\tau(x).$ This means
that $x\in(A\cap B)^{\diamond}.$ Thus, $A\cap B^{\diamond}\subseteq(A\cap
B)^{\diamond}.$ ∎

######  Definition 3.4.

Let $(X,\tau,\mathcal{P})$ be a primal topological space. We consider a map
$cl^{\diamond}:2^{X}\rightarrow 2^{X}$ as $cl^{\diamond}(A)=A\cup
A^{\diamond}$, where $A$ is any subset of $X$.

######  Theorem 3.5.

Let $(X,\tau,\mathcal{P})$ be a primal topological space and $A,B\subseteq X.$
Then, the following statements hold:

$(i)$ $cl^{\diamond}(\emptyset)=\emptyset$,

$(ii)$ $cl^{\diamond}(X)=X$,

$(iii)$ $A\subseteq cl^{\diamond}(A)$,

$(iv)$ If $A\subseteq B$, then $cl^{\diamond}(A)\subseteq cl^{\diamond}(B)$,

$(v)$ $cl^{\diamond}(A)\cup cl^{\diamond}(B)=cl^{\diamond}(A\cup B),$

$(vi)$ $cl^{\diamond}(cl^{\diamond}(A))=cl^{\diamond}(A).$

###### Proof.

Let $A,B\subseteq X.$

$(i)$ Since $cl^{\diamond}(\emptyset)=\emptyset,$ we have
$cl^{\diamond}(\emptyset)=\emptyset\cup\emptyset^{\diamond}=\emptyset.$

$(ii)$ Since $X\cup X^{\diamond}=X,$ we have $cl^{\diamond}(X)=X.$

$(iii)$ Since $cl^{\diamond}(A)=A\cup A^{\diamond},$ we have $A\subseteq
cl^{\diamond}(A).$

$(iv)$ Let $A\subseteq B.$ We get from (v) of Theorem 3.3 that
$A^{\diamond}\subseteq B^{\diamond}.$ Therefore, we have $A\cup
A^{\diamond}\subseteq B\cup B^{\diamond}$ which means that
$cl^{\diamond}(A)\subseteq cl^{\diamond}(B).$

$(v)$ It is obvious from the definition of the operator $cl^{\diamond}$ and
(v) of Theorem 3.3.

$(vi)$ It is obvious from (iii) that $cl^{\diamond}(A)\subseteq
cl^{\diamond}(cl^{\diamond}(A)).$ On the other hand, since $A^{\diamond}$ is
closed in $X,$ we have $(A^{\diamond})^{\diamond}\subseteq A^{\diamond}.$
Therefore,

$\begin{array}[]{rcl}cl^{\diamond}(cl^{\diamond}(A))&=&cl^{\diamond}(A)\cup\left(cl^{\diamond}(A)\right)^{\diamond}\\\
&=&cl^{\diamond}(A)\cup(A\cup A^{\diamond})^{\diamond}\\\
&=&cl^{\diamond}(A)\cup A^{\diamond}\cup(A^{\diamond})^{\diamond}\\\
&\subseteq&cl^{\diamond}(A)\cup A^{\diamond}\cup A^{\diamond}\\\
&=&cl^{\diamond}(A)\end{array}$

Thus, we have $cl^{\diamond}(cl^{\diamond}(A))=cl^{\diamond}(A).$ ∎

######  Corollary 3.5.1.

Let $(X,\tau,\mathcal{P})$ be a primal topological space. Then the function
$cl^{\diamond}:2^{X}\rightarrow 2^{X}$ defined by $cl^{\diamond}(A)=A\cup
A^{\diamond}$, where $A$ is any subset of $X,$ is a Kuratowski’s closure
operator.

######  Definition 3.5.

Let $(X,\tau,\mathcal{P})$ be a primal topological space. Then, the family
$\tau^{\diamond}=\\{A\subseteq X|cl^{\diamond}(A^{c})=A^{c}\\}$ is a topology
on $X$ induced by topology $\tau$ and primal $\mathcal{P}.$ It is called
primal topology on $X.$ We can also write $\tau_{\mathcal{P}}^{\diamond}$
instead of $\tau^{\diamond}$ to specify the primal as per our requirements.

######  Theorem 3.6.

Let $(X,\tau,\mathcal{P})$ be a primal topological space. Then the primal
topology $\tau^{\diamond}$ is finer than $\tau.$

###### Proof.

Let $A\in\tau.$ Then, $A^{c}$ is $\tau$-closed in $X.$ From (i) of Theorem
3.3, we get $(A^{c})^{\diamond}\subseteq A^{c}.$ Thus,
$cl^{\diamond}(A^{c})=A^{c}\cup(A^{c})^{\diamond}\subseteq A^{c}.$ Since
$A^{c}\subseteq cl^{\diamond}(A^{c})$ is always true for any subset $A$ of
$X,$ we obtain $cl^{\diamond}(A^{c})=A^{c}.$ This means that
$A\in\tau^{\diamond}.$ Thus, we have $\tau\subseteq\tau^{\diamond}.$ ∎

######  Theorem 3.7.

Let $(X,\tau,\mathcal{P})$ be a primal topological space. Then, the following
statements hold:

$(i)$ if $\mathcal{P}=\emptyset,$ then $\tau^{\diamond}=2^{X},$

$(ii)$ if $\mathcal{P}=2^{X}\setminus\\{X\\},$ then $\tau=\tau^{\diamond}.$

###### Proof.

$(i)$ We have always $\tau^{\diamond}\subseteq 2^{X}.$ Now, let $A\in 2^{X}.$
Since $\mathcal{P}=\emptyset,$ we have $A^{\diamond}=\emptyset$ for any subset
$A$ of $X.$ Therefore, $cl^{\diamond}(A^{c})=A^{c}.$ This means that
$A\in\tau^{\diamond}.$ Hence, $2^{X}\subseteq\tau^{\diamond}.$ Thus, we have
$\tau^{\diamond}=2^{X}.$
$(ii)$ We have always $\tau\subseteq\tau^{\diamond}$ from Theorem 3.6. Now, we
will prove that $\tau^{\diamond}\subseteq\tau.$ Let $A\in\tau^{\diamond}.$
Then $A^{c}\cup(A^{c})^{\diamond}=A^{c}$ which means that
$(A^{c})^{\diamond}\subseteq A^{c}.$ Now, let $x\notin(A^{c})^{\diamond}.$
Then, there exists $U\in\tau(x)$ such that $U^{c}\cup(A^{c})^{c}=U^{c}\cup
A\notin\mathcal{P}.$ Since $\mathcal{P}=2^{X}\setminus\\{X\\},$ we obtain
$U^{c}\cup A=X$ and so $U\cap A^{c}=\emptyset.$ Therefore, $x\notin
cl(A^{c}).$ Thus, we have $cl(A^{c})\subseteq(A^{c})^{\diamond}\subseteq
A^{c}.$ Hence, $cl(A^{c})=A^{c}$ which means that $A^{c}$ is $\tau$-closed and
so, $A\in\tau.$ Thus, $\tau^{\diamond}\subseteq\tau.$ Consequently, we have
$\tau=\tau^{\diamond}.$ ∎

######  Remark 3.5.

The converse of Theorem 3.7 need not be true as shown by the following
example.

######  Example 3.9.

Let $X=\\{a,b,c\\}$ with topology $\tau=\\{\emptyset,X,\\{a,b\\}\\}$ and
$\mathcal{P}=2^{X}\setminus\\{X,\\{a,b\\}\\}.$ Simple calculations show that
$\tau=\tau^{\diamond}$, but $\mathcal{P}$ is not equal to
$2^{X}\setminus\\{X\\}.$

######  Theorem 3.8.

Let $(X,\tau,\mathcal{P})$ be a primal topological space and $A\subseteq X.$
Then the followings hold:

$(i)$ $A\in\tau^{\diamond}$ if and only if for all $x$ in $A,$ there exists an
open set $U$ containing $x$ such that $U^{c}\cup A\notin\mathcal{P},$

$(ii)$ if $A\notin\mathcal{P},$ then $A\in\tau^{\diamond}.$

###### Proof.

$(i)$ Let $A\in\tau^{\diamond}.$

$\begin{array}[]{rcl}A\in\tau^{\diamond}&\Leftrightarrow&cl^{\diamond}(A^{c})=A^{c}\\\
&\Leftrightarrow&A^{c}\cup(A^{c})^{\diamond}=A^{c}\\\
&\Leftrightarrow&(A^{c})^{\diamond}\subseteq A^{c}\\\
&\Leftrightarrow&A\subseteq((A^{c})^{\diamond})^{c}\\\
&\Leftrightarrow&(\forall x\in A)(x\notin(A^{c})^{\diamond})\\\
&\Leftrightarrow&(\forall x\in A)(\exists
U\in\tau(x))(U^{c}\cup(A^{c})^{c}=U^{c}\cup A\notin\mathcal{P}).\end{array}$

$(ii)$ Let $A\notin\mathcal{P}$ and $x\in A.$ Put $U=X.$ Then, $U$ is a
$\tau$-open set containing $x.$ Since $A\notin\mathcal{P}$ and $U^{c}\cup
A=A,$ we have $U^{c}\cup A\notin\mathcal{P}.$ From (i), we get
$A\in\tau^{\diamond}.$ ∎

######  Theorem 3.9.

Let $(X,\tau,\mathcal{P})$ be a primal topological space. Then, the family
$\mathcal{B}_{\mathcal{P}}=\\{T\cap P|T\in\tau$ and $P\notin\mathcal{P}\\}$ is
a base for the primal topology $\tau^{\diamond}$ on $X.$

###### Proof.

Let $B\in\mathcal{B}_{\mathcal{P}}.$ Then, there exist $T\in\tau$ and
$P\notin\mathcal{P}$ such that $B=T\cap P.$ Since
$\tau\subseteq\tau^{\diamond},$ we get $T\in\tau^{\diamond}.$ On the other
hand, from Theorem 3.3 (ii), we have $P\in\tau^{\diamond}.$ Therefore,
$B\in\tau^{\diamond}.$ Consequently,
$\mathcal{B}_{\mathcal{P}}\subseteq\tau^{\diamond}.$ Now, let
$A\in\tau^{\diamond}$ and $x\in A.$ Then, from Theorem 3.8 (i), there exists
$U\in\tau(x)$ such that $U^{c}\cup A\notin\mathcal{P}.$ Now, let
$B=U\cap(U^{c}\cup A).$ Hence, we have $B\in\mathcal{B}_{\mathcal{P}}$ such
that $x\in B\subseteq A.$ ∎

######  Theorem 3.10.

Let $(X,\tau,\mathcal{P})$ and $(X,\tau,\mathcal{Q})$ be two primal
topological spaces. If $\mathcal{P}\subseteq\mathcal{Q},$ then
$\tau_{\mathcal{Q}}^{\diamond}\subseteq\tau_{\mathcal{P}}^{\diamond}.$

###### Proof.

Let $A\in\tau_{\mathcal{Q}}^{\diamond}.$ Then
$A^{c}\cup(A^{c})_{\mathcal{Q}}^{\diamond}=A^{c}$ which means that
$(A^{c})_{\mathcal{Q}}^{\diamond}\subseteq A^{c}.$ Now, let $x\notin A^{c}.$
Then, we get $x\notin(A^{c})_{\mathcal{Q}}^{\diamond}$ and so there exists
$U\in\tau(x)$ such that $U^{c}\cup(A^{c})^{c}=U^{c}\cup A\notin\mathcal{Q}.$
Since $\mathcal{P}\subseteq\mathcal{Q},$ we have $U^{c}\cup
A\notin\mathcal{P}.$ Therefore, $x\notin(A^{c})_{\mathcal{P}}^{\diamond}.$
Thus, $(A^{c})_{\mathcal{P}}^{\diamond}\subseteq A^{c}$ and so
$cl^{\diamond}(A^{c})=A^{c}\cup(A^{c})_{\mathcal{P}}^{\diamond}=A^{c}.$ Hence,
$A\in\tau_{\mathcal{P}}^{\diamond}.$ Consequently, we have
$\tau_{\mathcal{Q}}^{\diamond}\subseteq\tau_{\mathcal{P}}^{\diamond}.$ ∎

######  Theorem 3.11.

Let $f:X\to Y$ be a function and $\mathcal{P}\subseteq 2^{X}.$ If
$\mathcal{P}$ is a primal on $X$ and $f$ is not surjective, then
$\mathcal{Q}=\\{f(P)|P\in\mathcal{P}\\}$ is a primal on $Y$.

###### Proof.

$(i)$ Suppose that $Y\in\mathcal{Q}.$ Then there exists $P\in\mathcal{P}$ such
that $f(P)=Y.$ However, this contradicts the fact that $f$ is not surjective.

$(ii)$ Let $A\in\mathcal{Q}$ and $B\subseteq A.$ Then, there exists
$P_{1}\in\mathcal{P}$ such that $A=f(P_{1}).$ Now, set $P_{2}=f^{-1}(B)\cap
P_{1}.$ It is obvious that $P_{2}\subseteq P_{1}.$ Since $\mathcal{P}$ is a
primal on $X,$ $\mathcal{P}$ is downward closed and so we have
$P_{2}\in\mathcal{P}.$ Also, $B=f(P_{2}).$ This means that $B\in\mathcal{Q}.$

$(iii)$ Let $A\cap B\in\mathcal{Q}.$ Then, there exists
$P_{1},P_{2}\in\mathcal{P}$ such that $A=f(P_{1})$ and $B=f(P_{2}).$ Since
$\mathcal{P}$ is a primal on $X$ and $P_{1}\cap P_{2}\subseteq P_{1},$ we have
$P_{1}\cap P_{2}\in\mathcal{P}$ and so $P_{1}\in\mathcal{P}$ or
$P_{2}\in\mathcal{P}.$ Therefore, $A=f(P_{1})\in\mathcal{Q}$ or
$B=f(P_{2})\in\mathcal{Q}.$ ∎

######  Corollary 3.11.1.

It is obvious from Theorem 3.11 that the property of being primal is not a
topological property.

######  Remark 3.6.

For a function $f:X\to Y$ and a primal $\mathcal{Q}$ on $Y,$ the family
$\mathcal{P}=\\{f^{-1}(Q)|Q\in\mathcal{Q}\\}$ need not be a primal on $X$ as
shown by the following example.

######  Example 3.10.

Consider $X=\\{a,b\\},$ $Y=\\{1,2\\}$ and
$\mathcal{Q}=\\{\emptyset,\\{1\\}\\}.$ Define the function $f:X\to Y$ by
$f(x)=1.$ Then $\mathcal{Q}$ is a primal on $Y$ but
$\mathcal{P}=\\{f^{-1}(Q)|Q\in\mathcal{Q}\\}=\\{\emptyset,X\\}$ is not a
primal on $X.$

##  4 Conclusion

This paper introduced primal, which is a dual structure of grill. It is well
known that ideal, dual structure of filter, is one of the highly useful
notions in topology [21], summability theory [22], real analysis [23], etc.
Thus, ideal inspired us to introduce primal. Here, we introduced two new
operators using primal. One of these two operators satisfies Kuratowski’s
closure axioms. Moreover, we introduce a topology named primal topology
($\tau_{\mathcal{P}}$), which is finer than any topology $\tau$ of primal
topological space $(X,\tau,\mathcal{P})$. Later, we provided structure of base
for $\tau_{\mathcal{P}}$ and proved several fundamental results. It is
important to note that primal structures and some related results showed
quantum behaviours i.e. we could not determine some properties as universally
true. For example, one may refer examples 3.6 and 3.7. Moreover, Corollary
3.11.1 and Remark 3.6 showed some uncertain behaviours of this new structure.
Thus, we hope that we should study this new notion more deeply in general
topology and other areas. If possible, we are looking forward to connect this
notion with some ideas of quantum world [24] from the perspective of general
topology in future.

Conflict of interest: The authors declare that there is no conflict of
interest.

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