# Systematically Improvable Generalization of Self-Interaction Corrected
Density Functional Theory

Benjamin G. Janesko  Department of Chemistry & Biochemistry, Texas Christian
University, 2800 S. University Dr, Fort Worth, TX 7629, USA

###### Abstract

Perdew-Zunger self-interaction correction (PZSIC) reintroduces an exact
constraint to approximate density functional theory (DFT), but can
paradoxically degrade performance and is not systematically improvable. We use
the Adiabatic Projection formalism to derive PZSIC in terms of a reference
system experiencing only electron self-interaction. Generalization introduces
correlation into the reference system, systematically bridging from PZSIC to
exact wavefunction theory. Minimal active spaces resolve the PZSIC paradox,
accurately treating near-equilibrium and strongly-correlated systems.

Kohn-Sham density functional theory is the most widely-used electronic
structure approximation. KS-DFT models a real system of interacting electrons
in terms of a reference system of noninteracting Fermions, corrected by mean-
field (Hartree) electron repulsion and a formally exact exchange-correlation
(XC) spin-density functional. Standard approximate XC functionals typically
work well for properties such as lattice constants and bond energies, but can
fail dramatically for properties such as band gaps, some reaction barriers,
charge-transfer excited states, and stretched bonds.Perdew2009 ; Zope2019
Failures typically involve (1) nearly-one-electron regions where approximate
XC functionals do not cancel the spurious Hartree self-interaction, and (2)
strongly correlated regions where the exact reference system wavefunction
substantially differs from the exact real system wavefunction.Janesko2021

The Perdew-Zunger self-interaction correctionPerdew1981  addresses failure (1)
by removing self-interaction error in an orbital-by-orbital fashion,

$\displaystyle E_{XC}^{SICSL}[\rho_{\uparrow},\rho_{\downarrow}]$
$\displaystyle=$ $\displaystyle
E_{XC}^{SL}[\rho_{\uparrow},\rho_{\downarrow}]-\sum_{i,\sigma}^{occ}U[\rho_{i\sigma}]+E_{XC}^{SL}[\rho_{i\sigma},0].$
(1)

$\rho_{\sigma}({\bf{r}})=\sum_{i=1}^{N\sigma}|\psi_{i\sigma}({\bf{r}})|^{2}$
is the probability density of the $N_{\sigma}$
$\sigma=\uparrow,\downarrow$-spin electrons, $E_{XC}^{SL}$ is a
semilocalBurke1998  approximation to the exact XC functional, self-Hartree
energy $U[\rho_{i\sigma}]=(1/2)\int d^{3}{\bf{r}}_{1}\int
d^{3}{\bf{r}}_{2}\rho_{i\sigma}({\bf{r}}_{1})|{\bf{r}}_{1}-{\bf{r}}_{2}|^{-1}\rho_{i\sigma}({\bf{r}}_{2})$,
and the reference system’s single Slater determinant wavefunction is composed
of spin-orbitals $\\{\psi_{i\sigma}\\}$. Orbital densities
$\rho_{i\sigma}=|\phi_{i\sigma}|^{2}$ are constructed from localized orbitals
$\\{\phi_{i\sigma}\\}$ defined as a unitary transform of
$\\{\psi_{i\sigma}\\}$. While optimizing this transform can be
expensive,Pederson1984 ; Vydrov2004  complexJonsson2011  or Fermi-
LöwdinPederson2014  orbitals have made SIC increasingly practical.Sharkas2020

This has led to a renewed focus on the ”paradox” of self-interaction.
Semilocal functionals’ self-interaction simulates some opposite-spin
correlation, thus PZSIC can degrade performance for systems near equilibrium
and especially for strongly correlated systems.Zope2019  This exemplifies the
broader ”zero-sum” tradeoffs of standard XC approximations.Janesko2021  One
road beyond this tradeoff is to reintroduce part of the electron-electron
interaction into the reference system.Savin1988 ; Fromager2007

Here we address the ”paradox” of self-interaction by first explicitly deriving
the PZSIC ansatz to address failure (1), then generalizing to address failure
(2). The derivation uses the Adiabatic Projection formalism.Janesko2022a
Consider $N$ electrons in external potential $v_{ext}({\bf{r}})$. Define
$\hat{T}$ and $\hat{V}_{ee}$ as the real system’s kinetic and electron-
electron repulsion operators. Define
$\hat{P}_{i\sigma}=\left|\phi_{i\sigma}\phi_{i\sigma}\right>\left<\phi_{i\sigma}\phi_{i\sigma}\right|$
as a two-electron projection onto localized orbital $\phi_{i\sigma}$. (This
denotes the direct product:
$\left<{\bf{r}}_{1},{\bf{r}}_{2}|\phi_{i\sigma}\phi_{i\sigma}\right>=\phi_{i\sigma}({\bf{r}}_{1})\phi_{i\sigma}({\bf{r}}_{2})$.)
Define a projected operator
$\hat{V}^{P}_{ee}=\sum_{i\sigma}\hat{P}_{i\sigma}\hat{V}_{ee}\hat{P}_{i\sigma}$.
Introduce an adiabatic connection parameter $\lambda$ and a generalized energy
functionalLevy1985a

$\displaystyle E[\lambda,\\{\hat{P}_{i\sigma}\\},\rho]$ $\displaystyle=$
$\displaystyle\int
d^{3}{\bf{r}}\rho({\bf{r}})v_{ext}({\bf{r}})+\min_{\Psi\to\rho}\left<\Psi\right|\hat{T}+\hat{V}^{P}_{ee}+\lambda(\hat{V}_{ee}-\hat{V}^{P}_{ee})\left|\Psi\right>$
(2)

Here $\lambda=1$ is the real system and $\lambda=0$ is a reference system
experiencing only the projected electron-electron repulsion operator. Define
$\Psi[\lambda,\\{\hat{P}_{i\sigma}\\},\rho]$ as the minimizing wavefunction of
Eq. 2. The reference system experiences only electron self-interaction, thus
its electron-electron interaction energy is zero and its ground-state
wavefunction $\Phi[\lambda=0,\\{\hat{P}_{i\sigma}\\},\rho]$ is a single Slater
determinant composed of the spin-orbitals $\\{\psi_{i\sigma}\\}$ introduced
above. Explicitly,

$\displaystyle E_{ee}[\lambda=0,\\{\hat{P}_{i\sigma}\\},\rho]$
$\displaystyle=$
$\displaystyle\left<\Phi[\lambda=0,\\{\hat{P}_{i\sigma}\\},\rho]|\sum_{i\sigma}\hat{P}_{i\sigma}\hat{V}_{ee}\hat{P}_{i\sigma}|\Phi[\lambda=0,\\{\hat{P}_{i\sigma}\\},\rho]\right>$
$\displaystyle=$ $\displaystyle\sum_{i\sigma
jk}c^{*}_{ji}c^{*}_{ki}U[\rho_{i\sigma}]\left(c_{ij}c_{ik}-c_{ik}c_{ij}\right)\
=\ \sum_{i\sigma}U[\rho_{i\sigma}]+E_{X}^{ex}[\rho_{i\sigma}]$

Here $c_{ij}=\left<\phi_{i\sigma}|\psi_{j\sigma^{\prime}}\right>$ ensuring
$\left(c_{ij}c_{ik}-c_{ik}c_{ij}\right)$ is zero. Self-Hartree energy
$U[\rho_{i\sigma}]=\left<\phi_{i\sigma}\phi_{i\sigma}|\hat{V}_{ee}|\phi_{i\sigma}\phi_{i\sigma}\right>$
was introduced above. Self-exchange
$E_{X}^{ex}[\rho_{i\sigma}]=-U[\rho_{i\sigma}]$. In PZSIC, localized orbitals
$\\{\phi_{i\sigma}\\}$ are a unitary transform of $\\{\psi_{i\sigma}\\}$
ensuring $\sum_{j}c^{*}_{ji}c_{ij}=1$. The Hohenberg-Kohn theorems ensure that
the real system’s ground-state density and energy can be obtained by
minimizing the energy of the reference system, corrected by some density
functional

$\displaystyle E_{HXC}[\\{\hat{P}_{i\sigma}\\},\rho]$ $\displaystyle=$
$\displaystyle\int_{0}^{1}d\lambda\left<\Psi[\lambda,\\{\hat{P}_{i\sigma}\\},\rho]\right|\hat{V}_{ee}-\hat{V}^{P}_{ee}\left|\Psi[\lambda,\\{\hat{P}_{i\sigma}\\},\rho]\right>$
$\displaystyle=$ $\displaystyle
U[\rho]-\left(\sum_{i\sigma}U[\rho_{i\sigma}]\right)+E_{XC}[\\{\hat{P}_{i\sigma}\\},\rho]$

Ref. Janesko2022a  gives details of this adiabatic connetion. The $U[\rho]$
and $U[\rho_{i\sigma}]$ terms in eq Systematically Improvable Generalization
of Self-Interaction Corrected Density Functional Theory are respectively from
the $\lambda=1$ and $\lambda=0$ limits of the adiabatic connection.
Approximating the projected XC functional as
$E_{XC}[\\{\hat{P}_{i\sigma}\\},\rho]\simeq
E_{XC}^{SL}[\rho_{\uparrow},\rho_{\downarrow}]-\sum_{i\sigma}E_{XC}^{SL}[\rho_{i\sigma},0]$
recovers the PZSIC.

With this derivation in hand, generalization of PZSIC to treat failure (2)
merely requires generalizing the projections from single orbitals to active
spaces. We illustrate a closed-shell $N_{\sigma}$-electron system such as
stretched singlet Li2 (Figure 1) or twisted 2-butene (Figure 2), with a single
active space of two electrons in two orbitals (four spin-orbitals)
$\psi_{h\sigma}$ and $\psi_{l\sigma}$. Choose $\\{\phi_{i\sigma}\\}$ as a
unitary transform of the $N_{\sigma}-1$ remaining ”core” orbitals. Define a
projection onto the active space and a projected electron-electron repulsion
operator,

$\displaystyle\hat{P}_{22}$ $\displaystyle=$
$\displaystyle\sum_{a,b=h,l}\sum_{\sigma,\sigma^{\prime}}\left|\psi_{a\sigma}\psi_{b\sigma^{\prime}}\right>\left<\psi_{a\sigma}\psi_{b\sigma^{\prime}}\right|,$
(5) $\displaystyle\hat{V}^{P22}_{ee}$ $\displaystyle=$
$\displaystyle\hat{P}_{22}\hat{V}_{ee}\hat{P}_{22}+\sum_{\sigma}\sum_{i=1}^{N\sigma-1}\hat{P}_{i\sigma}\hat{V}_{ee}\hat{P}_{i\sigma}.$
(6)

The reference system’s electron-electron interaction energy is now nonzero,
and its exact ground state $\Psi^{CAS}[\lambda=0,\rho]$ is a multireference
complete active space CAS(2,2) wavefunction capable of giving noninteger
active orbital occupancies $n_{h\sigma},n_{l\sigma}$. The real system’s
electron-electron interaction energy is modeled as

$\displaystyle E_{ee}$ $\displaystyle=$
$\displaystyle\left<\Psi^{CAS}[\lambda=0,\rho]|\hat{V}_{ee}|\Psi^{CAS}[\lambda=0,\rho]\right>+(E_{XC}^{SL}[\rho]-E_{XC}^{SL}[\rho_{a}])$
$\displaystyle-(E_{X}^{ex}[\rho]-E_{X}^{ex}[\rho_{a}])-\sum_{\sigma}\sum_{i\sigma=1}^{N\sigma-1}\left(U[\rho_{i\sigma}]+E_{XC}^{SL}[\rho_{i\sigma},0]\right)$

Here $\rho_{a}=n_{h}|\psi_{h}|^{2}+n_{l}|\psi_{l}|^{2}$ is the electron
density projected onto the active space. The first expectation value in Eq
Systematically Improvable Generalization of Self-Interaction Corrected Density
Functional Theory includes the Hartree energy $U[\rho]$, multireference
exchange and correlation in the active space, and exact exchange outside the
active space. The other terms add semilocal XC outside the active space,
subtract off exact exchange outside the active space, and apply the SIC to the
remaining core orbitals. The approach is systematically improvable in the way
that correlated wavefunction theory is systematically improvable, recovering
full configuration interaction (FCI) in the limit where the active space
projector includes all occupied and unoccupied orbitals. It explicitly avoids
”double-counting” of exchange-correlation, just as PZSIC avoids ”double-
counting” of exchange-correlation. We use ”APSIC($n,m$)SL” to denote
generalized self-interaction correction with a single active space of $n$
electrons and $m$ orbitals, using semilocal functional ”SL”. ”APCAS($n,m$)SL”
denotes omitting the core-orbital SIC (the sums over $i\sigma$ in Eq.
Systematically Improvable Generalization of Self-Interaction Corrected Density
Functional Theory).

Figure 1:  Dissociation curves of Li${}_{2}^{+}$ (top), spin-restricted
singlet Li2 (middle), and spin-restricted singlet N2 (bottom), dissociation
energies relative to separate atoms.

Figure 1 shows APSIC($n,2$)PBE dissociation curves of Li${}_{2}^{+}$ ($n=1$),
a system where PZSIC fixes failure (1), and spin-restricted singlet Li2
($n=2$), a system where PZSIC worsens failure (2). Calculations use the
def2-TZVP basis set,Weigend2005  the PySCF package,Sun2017 ; Sun2020  the
Perdew-Burke-Ernzerhof (PBE) semilocal XC functional,Perdew1996  CAS($n,m$)SCF
canonical orbitals, and Edmiston-Ruedenberg localized core
orbitalsEdmiston1963  to compute the core-orbital SIC. Dissociation energies
are relative to separated atoms, evaluated with $n=0$. ”Ref” denotes
N-electron valence perturbation theory with the CAS($n,m$)SCF
reference.Angeli2001  Code to perform all calculations is freely available at
github.org/bjanesko/apsic.

The generalized SIC fixes both failures of standard DFT. APSIC($1,2$)PBE
recovers PZSIC for Li${}_{2}^{+}$, fixing the overbinding of PBE (failure
(1)). APSIC($2,2)$PBE gives a near-exact dissociation limit for strongly
correlated spin-restricted Li2 (failure (2)), and is close to the reference
curve near equilibrium.

The 3.6 kcal/mol APSIC(2,2)PBE overbinding of dissociated Li2 arises not from
”double-counting”, but from PBE overbinding outside of the active space. PBE
(HF) calculations on stretched Li2 give an XC contribution to the bond energy
$E_{XC}(Li_{2})-2E_{XC}(Li)$ of +13.6 (+73.4) kcal/mol, with +17.2 (+73.4)
kcal/mol from the active space, leaving a difference of 3.6 (0.0) kcal/mol.
PBE ”overbinds” the dissociation limit relative to HF theory, and not all of
that overbinding is within the active space. Correcting the active space with
wavefunction correlation leaves a residual overbinding. The bottom panel of
Figure 1 shows similar effects for the stretched singlet symmetry-restricted
N2 triple bond. APSIC(6,6)PBE overbinding can be fixed by increasing the
active space, or with the APCAS(6,6)PBEH ”half-and-half” global
hybrid.Becke1993  Pair density functionals address related effects in other
wavefunction-in-DFT methods.Sharma2021 ; Perdew1995

Figure 2 shows the APSIC(2,2)PBE torsional potential energy surface for
2-butene double bond rotation. Calculations use the 6-31G(d) basis set and
spin-unrestricted UB3LYP/6-31G* geometries optimized in Gaussian 16.g16
Hartree-Fock, PBE, and PZSIC-PBE calculations give an unphysical cusp due to
failure (2). APSIC(2,2)PBE is instead quite close to the reference.

Figure 2:  Torsional potential energy surface for 2-butene double bond
rotation.

These results suggest a number of avenues for future work. (1) The Adiabatic
Projection derivation admits many choices of projection $\hat{P}_{i\sigma}$,
including the complex-valuedLehtola2016  or Fermi-LöwdinPederson2014
localized orbitals of modern SIC. This derivation works with those methods.
Moreover, there is no requirement to choose the localized orbitals as a
unitary transform of the canonical orbitals. Other choices, such as predefined
atomic orbitals, may provide formal connections to methods such as
DFT+U.Kulik2015  (2) The present approximation does not include active-virtual
correlation. APSIC(2,2)PBE calculations on a closed-shell two-electron system
recover CAS(2,2), not FCI. (Of course, increasing the active space is
guaranteed to converge to FCI.) Active-virtual correlations could be added
using existing wavefunctionChatterjee2016  or wavefunction-in-
DFTGraefenstein2005  approaches. (3) Introducing multiple orthogonal active
spaces provides connections to other correlated wavefunction model
chemistries. For example, while treating an $N$-electron system with $N$
mutually orthogonal one-orbital spaces recovers PZSIC, $N$ mutually orthogonal
two-orbital spaces could recover a ”DFT-corrected” version of perfect pairing
/ antisymmetrized geminal product approximations.Beran2005 ; Pernal2013
Localization of the active space orbitals provides other connections to
linear-scaling localized-orbital ab initio methods.Scuseria1999  Overall, the
results motivate further application of the Adiabatic Projection formalism to
computationally tractable model chemistries beyond standard DFT.

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# Almost separable spaces

Sagarmoy Bag  Department of Pure Mathematics, University of Calcutta, 35,
Ballygunge Circular Road, Kolkata 700019, West Bengal, India
<EMAIL_ADDRESS>,  Ram Chandra Manna  Ramakrishna Mission
Vidyamandir, Belur Math, Howrah-711202, West Bengal, India
<EMAIL_ADDRESS>and  Sourav Kanti Patra  Ramakrishna Mission
Vidyamandir, Belur Math, Howrah-711202, West Bengal, India
<EMAIL_ADDRESS>
###### Key words and phrases:

Almost dense set, Almost separable space, Pseudocompact space, functionally
Hausdorff space, sequentially separabile space, Strongly sequentially
separabile space

###### 2010 Mathematics Subject Classification:

Primary 54A05; Secondary 54D65

## Abstract

We have defined almost separable space. We show that like separability, almost
separability is $c$ productive and converse also true under some restrictions.
We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts
spaces. We investigate few relationships among separability, almost
separability, sequential separability, strongly sequential separability.

##  1\. Introduction

Let $X$ be any topological space. $C(X)$ be the set of all real valued
continuous functions on $X$. A subset $A$ of $X$ is called almost dense in $X$
if for any $f\in C(X)$ with $f(A)=\\{0\\}$, implies that $f(X)=\\{0\\}$. And a
topological space $X$ is called almost separable if it has a countable almost
dense subset. A dense subset is always almost dense. In completely regular
space, dense and almost dense sets are identical. But the converse is not
true. We give an example of a non completely regular space in which dense and
almost dense sets are same [ Example 2.5 ]. Theorem 3.5 shows that almost
separability is $c$ productive and under some restrictions the converse is
also true Theorem 3.8. In theorem 4.3, we established Baire Category like
theorem. Finally we establish relationships among almost separability,
sequentially separability and strongly sequentially separability.

######  Definition 1.1.

[1] A space $X$ is called sequentially separable if there exist a countable
set $D$ such that for every $x\in X$ there exist a sequence from $D$
converging to $x$

######  Definition 1.2.

[1] A space $X$ is called strongly sequentially separable if it is separable
and every dense countable subspace is sequentially dense. A subset $D$ of $X$
is called sequentially dense if for every $x$ in $X$ there exists a sequence
from $D$ converges to $x$.

##  2\. Almost dense subsets

######  Definition 2.1.

A subset $A$ of a topological space $X$ is called almost dense if for any
$f\in C(X)$ with $f(A)=\\{0\\}$, implies $f(X)=\\{0\\}$.

######  Theorem 2.2.

Dense subsets are always almost dense.

###### Proof.

Obvious. ∎

######  Theorem 2.3.

Let $X$ be a completely regular space, then almost dense set are dense in $X$.

###### Proof.

Let $A$ be almost dense subset of $X$. If possible let $A$ be not dense in $X$
and $x_{\circ}\in X\setminus\overline{A}$ then there exists $f\in C(X)$ such
that $f(x_{\circ})=0$ and $f(\overline{A})=1$. Then $f(A)=0$ but
$f(X)\neq\\{0\\}$. This implies that $A$ is not almost dense in $X$ which is a
contradiction. Hence $A$ is dense in $X$. ∎

We now show that the converse of the above theorem is not true.

######  Theorem 2.4.

Let $Y$ be an open dense subset of a topological space $X$ (may or may not be
a Completely regular space). If $Y$ has a base of cozero subsets of $X$. Then
almost dense set in $X$ is dense in $X$. A set $A$ of $X$ is called cozero set
if $A=X\setminus Z(f)$, for some $f\in C(X)$, where $Z(f)=\\{x\in
X:f(x)=0\\}$.

###### Proof.

Let $A$ be almost dense subset of $X$. To show $A$ is dense in $X$. Suppose
$A\cap Y=\emptyset$. Let $y\in Y$. Then there exist $f\in C(X)$ such that
$y\in X\setminus Z(f)\subseteq Y$. Then $f(A)=\\{0\\}$ but $f(X)\neq\\{0\\}$,
which is a contradiction as $A$ is almost dense in $X$. So, $A\cap Y\neq\phi$.
Claim: $A\cap Y$ is dense in $Y$. If not there is a cozero set $X\setminus
Z(g)\subset Y$ such that $(A\cap Y)\cap(X\setminus Z(g))=$, where $g\in C(X)$.
Then $Z(g)\subset(X\setminus Y)\cup(A\cap Y)\supset A$. So, that
$g(A)=\\{0\\}$. But $g(X)\neq\\{0\\}$. Contradiction shows that $A\cap Y$ is
d4ense in $Y$. Given that $Y$ is dense in $X$, $A\cap Y$ is dense in $Y$ which
yields $A$ is dense in $X$. ∎

To find such a spaces we present following examples.

######  Example 2.5.

Let $K=\\{\frac{1}{n}:n\in\mathbb{N}\\}$ and
$\beta=\\{(a,b):a<b,a,b\in\mathbb{R}\\}\cup\\{(a,b)\setminus
K:a<b,a,b\in\mathbb{R}\\}$ be a base for the topology $\tau_{K}$ on
$\mathbb{R}$ which is known as $K$-topology on $\mathbb{R}$. Then
$(\mathbb{R},\tau_{K})$ is not regular and hence not completely regular. We
would like to show that almost dense set in $(\mathbb{R},\tau_{K})$ is dense
in $(\mathbb{R},\tau_{K})$. Let $Y=\mathbb{R}\setminus\\{0\\}$. The set $Y$ is
open, dense in $(\mathbb{R},\tau_{K})$ and the relative topology of $Y$ is
usual topology of $Y$ which is completely regular and , as a result, possess a
base consisting of cozero sets of $(\mathbb{R},\tau_{K})$. We invoke theorem
2.4 to conclude that almost dense set in $(\mathbb{R},\tau_{K})$ are dense in
$(\mathbb{R},\tau_{K})$.

######  Example 2.6.

Let $X$ be zero-dimensional, non normal space. Let $F,G$ be two disjoint
closed sets in $X$ which can not be separated by disjoint open sets. Then
define $Y$ to be the quotient space obtaining by collapsing $G$ to a single
point.

The following example shows that in a normal space the almost dense set and
dense set may not be same.

######  Example 2.7.

Let $X=\\{a,b\\}$ and $\tau=\\{\phi,X,\\{a\\}\\}$ be a topology on $X$. Then
$X$ is a normal space. $\\{b\\}$ is almost dense set in $X$ but not dense in
$X$.

######  Theorem 2.8.

Let $f:X\mapsto Y$ be a continuous onto function and $A$ be almost dense set
in $X$, then $f(A)$ is almost dense set in $Y$.

###### Proof.

Let $g\in C(Y)$ such that $g(f(A))=\\{0\\}$. Then $g\circ f\in C(X)$ such that
$g\circ f(A)=\\{0\\}$. As $A$ is almost dense in $X$, then we have $g\circ
f(X)=\\{0\\}$. Since $f$ is onto we have $g(Y)=\\{0\\}$. Hence $f(A)$ is
almost dense in $Y$. ∎

######  Theorem 2.9.

Let $\tau_{1},\tau_{2}$ be two topologies on $X$ such that $\tau_{2}$ is finer
than $\tau_{1}$. Then if a subset $A$ of $X$ is almost dense in
$(X,\tau_{2})$, then $A$ is almost dense in $(X,\tau_{1})$.

###### Proof.

We denote $C(X,\tau_{i})$ for the set of all real valued continuous functions
on $X$ with respect to the topology $\tau_{i}$ for $i=1,2$. Let $A$ be almost
dense in $(X,\tau_{2})$. Let $f\in C(X,\tau_{1})$ with $f(A)=\\{0\\}$. Since
$f\in C(X,\tau_{2})$ as $\tau_{2}$ is finer than $\tau_{1}$. Since $A$ is
almost dense in $(X,\tau_{2})$, then $f(X)=\\{0\\}$. Hence $A$ is almost dense
in $(X,\tau_{1})$. ∎

######  Theorem 2.10.

If $A$ is almost dense in $X$ and $B$ is almost dense in $Y$, then $A\times B$
is almost dense in $X\times Y$.

###### Proof.

Let $f:X\times Y\mapsto\mathbb{R}$ be continuous such that $f=0$ on $A\times
B$. Fix $a\in A$ and look at $f_{a}(y)=f(a,y),y\in Y$.
$f_{a}:Y\mapsto\mathbb{R}$ is continuous and $f_{a}(y)=0$ for all $y\in B$. By
hypothesis $f_{a}(y)=0$ for all $y\in Y$.

Since $a$ is arbitrary we obtain $f(A\times Y)=\\{0\\}$. Take any $(x,y)\in
X\times Y$. then $f_{y}:X\mapsto\mathbb{R}$ defined by $f_{y}(z)=f(z,y),z\in
X$ is a continuous map. Now $f_{y}(a)=0$ for all $a\in A$. Then $f_{y}(x)=0$
for all $x\in X$ i.e. $f(x,y)=0$. Since $y\in Y$ is arbitrary this shows $f=0$
on $X\times Y$. Hence $A\times B$ is almost dense in $X\times Y$. ∎

######  Corollary 2.11.

Let $A_{1},A_{2},...,A_{n}$ be almost dense in $X_{1},X_{2},....,X_{n}$
respectively. Then $\prod_{i=1}^{n}A_{i}$ is almost dense in
$\prod_{i=1}^{n}X_{i}$.

######  Proposition 2.12.

Let $\\{X_{\alpha}:\alpha\in\Lambda\\}$ be a family of topological spaces and
$A_{\alpha}\subset X_{\alpha},\alpha\in\Lambda$ be almost dense. Then
$\prod_{\alpha\in\Lambda}A_{\alpha}$ is almost dense in
$\prod_{\alpha\in\Lambda}X_{\alpha}$.

###### Proof.

For $a=(a_{\alpha})_{\alpha\in\Lambda}\in\prod_{\alpha\in\Lambda}A_{\alpha}$.
Define
$D=\\{(x_{\alpha})_{\alpha\in\Lambda}\in\prod_{\alpha\in\Lambda}X_{\alpha}:\\{\alpha:x_{\alpha}\neq
a_{\alpha}\\}$ is finite$\\}$. Let
$f:\prod_{\alpha\in\Lambda}X_{\alpha}\mapsto\mathbb{R}$ be continuous such
that $f=0$ in $\prod_{\alpha\in\Lambda}A_{\alpha}$. For any finite subset
$I\subset\Lambda,\prod_{\alpha\in
I}X_{\alpha}\times\\{a_{\alpha}:\alpha\in\Lambda\setminus I\\}$ is
homeomorphic to $\prod_{\alpha\in I}X_{\alpha}$ and $\prod_{\alpha\in
I}A_{\alpha}$ is almost dense in $\prod_{\alpha\in I}X_{\alpha}$(by above
Corollary). Hence $\prod_{\alpha\in
I}A_{\alpha}\times\\{a_{\alpha}:\alpha\in\Lambda\setminus I\\}$ is almost
dense in $\prod_{\alpha\in
I}X_{\alpha}\times\\{a_{\alpha}:\alpha\in\Lambda\setminus I\\}$. Consequently
$f(\prod_{\alpha\in\Lambda}X_{\alpha}\times\\{a_{\alpha}:\alpha\in\Lambda\setminus
I\\})=\\{0\\}$. Now $D=\cup[\prod_{\alpha\in
I}X_{\alpha}\times\\{a_{\alpha}:\alpha\in\Lambda\setminus
I\\}:I\subset\Lambda$ finite $]$. This implies that $f(D)=\\{0\\}$. We now
invoke the following important result in product space. ∎

We now invoke the following important result on product space.

######  Theorem 2.13.

Let $\\{Y_{\alpha}:\alpha\in\Lambda\\}$ be a family of topological spaces and
and $a=(x_{\alpha})_{\alpha\in\Lambda}$ be a fixed element of
$\prod_{\alpha\in\Lambda}Y_{\alpha}$. The set
$E=\\{(y_{\alpha})_{\alpha\in\Lambda}\in\prod_{\alpha\in\Lambda}Y_{\alpha}:\\{\alpha\in\Lambda:y_{\alpha}\neq
x_{\alpha}\\}$ is finite$\\}$ is dense in
$\prod_{\alpha\in\Lambda}Y_{\alpha}$.

Because of this theorem $D$ is dense in $\prod_{\alpha\in\Lambda}X_{\alpha}$
and $f(D)=0$ which implies $f=0$ on $\prod_{\alpha\in\Lambda}X_{\alpha}$. As a
result $\prod_{\alpha\in\Lambda}A_{\alpha}$ is almost dense in
$\prod_{\alpha\in\Lambda}X_{\alpha}$.

######  Theorem 2.14.

If a topological space $X$ contains a connected almost dense subset, then $X$
is connected.

###### Proof.

Let $A$ be connected almost dense subset of $X$. If possible let $X$ be
disconnected. Then there exists a onto continuous function
$f:X\mapsto\\{0,1\\}$. Let $Y=\\{x\in X:f(x)=0\\}$. Then $X\setminus
Y=f^{-1}\\{1\\}$. Since $A$ is connected, then either $A\subseteq Y$ or
$A\subseteq X\setminus Y$.

Case 1: If $A\subseteq Y$. Then $f(A)=\\{0\\}$ but $f(X)\neq\\{0\\}$, which
contradicts the fact that $A$ is almost dense set in $X$.

Case 2: If $A\subseteq X\setminus Y$. Consider $g=1-f$. Then $g(A)=\\{0\\}$
but $g(X)\neq\\{0\\}$, which is a contradiction as $A$ is almost dense set in
$X$.

Hence $X$ is connected. ∎

Of course the converse of 2.14 is not true i.e., $X$ is a connected space but
an almost dense subset need not be connected. As an illustration consider the
following:

$X=\mathbb{R}$ and the topology consist of all subsets of $\mathbb{R}$
containing $0$. $X$ is obviously connected. The subspace $A=\\{1,2\\}$ is
almost dense in $X$ but not connected.

######  Theorem 2.15.

A subset $A$ of $X$ is almost dense in $X$ if and only if every non empty
cozero set intersects $A$.

###### Proof.

Let $A$ be an almost dense subset of $X$ and $X-Z(f)$ be a non-empty cozero
set in $X$. Let us assume that $A\cap(X-Z(f))=\emptyset$. Then $f(A)=\\{0\\}$.
Since $A$ is almost dense in $X$ , then $f(X)=\\{0\\}$ ,Which contradicts the
fact that $X-Z(f)$ is non-empty. So $A\cap\ (X-Z(f))\neq\emptyset$.

Conversely, let $A$ intersect every non-empty cozero sets of $X$. Suppose $A$
is not almost dense subset of $X$. Then there exists a continuous function
$f:X\mapsto\mathbb{R}$ such that $f(A)=\\{0\\}$ but $f(X)\neq\\{0\\}$. Then
$X-Z(f)$ is a non empty cozero set and $A$ does not intersect the non-empty
cozero set $X-Z(f)$. ∎

As is well known that in a topological space a non-empty subset is dense if
and only if it intersect every non-empty open set. Theorem 2.15 presents the
analogous result in the case of almost denseness property. We have seen that
in a topological space $X$ if $A\subset X$ is dense and $U$ is non-empty open
set, then not only $A\cap U\neq\emptyset$, $A\cap U$ is dense in $U$ also. Now
the question arises as follows:

Let $A$ be almost dense in a topological space $X$ and $\emptyset\neq U\subset
X$ is a cozero set. Then $A\cap U\neq\emptyset$ is no doubt, but $A\cap U$ is
almost dense in $U$? The answer does not seem to be clear!

##  3\. Almost separable spaces

######  Definition 3.1.

A space $X$ is called almost separable if it contains a countable almost dense
subset.

######  Theorem 3.2.

Each separable space is almost separable.

The converse of the above theorem is not true.

######  Example 3.3.

Let $X=\mathbb{R}$ and $\tau_{c}$ be cocountable topology on $X$. For any
countable set $A$ in $X$, $X\setminus A$ is open. Therefore $A$ is not dense
in $X$. Therefore $X$ is not separable.

We want to show that $\mathbb{Q}$ is almost dense in $X$. Let $f\in C(X)$ and
$f(\mathbb{Q})=\\{0\\}$. Since each real valued continuous function on $X$ is
constant, then $f(X)=\\{0\\}$. Therefore $\mathbb{Q}$ is almost dense in $X$.
Therefore $X$ contains a countable almost dense subset. Hence $X$ is an almost
separable space. In fact, any countable subset of $X$ is almost dense in $X$.

######  Theorem 3.4.

Finite product of almost separable spaces is almost separable.

###### Proof.

It follows from the Corollary 2.11. ∎

In the case of infinite products, like the case of separable spaces, the
following result is true.

######  Theorem 3.5.

Let $\\{X_{\alpha}:\alpha\in\Lambda\\}$ be a family of almost separable
topological spaces, $\Lambda$, with card($\Lambda)=c$, Then
$\prod_{\alpha\in\Lambda}X_{\alpha}$ with product topology is almost separable
space.

###### Proof.

Let $A_{\alpha}\subset X_{\alpha}$ be a countable almost dense subset of
$X_{\alpha}$. To avoid triviality we assume $A_{\alpha}$ to be countably
infinite. Let $f_{\alpha}:\mathbb{N}\mapsto A_{\alpha}$ be a bijection. Define
$\prod_{\alpha\in\Lambda}f_{\alpha}:\mathbb{N}^{\Lambda}\mapsto\prod_{\alpha\in\Lambda}A_{\alpha}$
as follows:
$\prod_{\alpha\in\Lambda}f_{\alpha}{(n_{\alpha},\alpha\in\Lambda)}=(f_{\alpha}(n_{\alpha}),\alpha\in\Lambda)$
where $(n_{\alpha},\alpha\in\Lambda)\in\mathbb{N}^{\Lambda}$. We write
$f^{\Lambda}$ for $\prod_{\alpha\in\Lambda}f_{\alpha}$. We know
$\mathbb{N}^{\Lambda}=\\{g:\Lambda\mapsto\mathbb{N}\\}$ and
$f^{\Lambda}(g)=(f_{\alpha}(g(\alpha)):\alpha\in\Lambda)\in\prod_{\alpha\in\Lambda}A_{\alpha}$.
It is easy to see that $f^{\Lambda}$ is onto. Let
$p_{\alpha}:\prod_{\alpha\in\Lambda}A_{\alpha}\mapsto A_{\alpha}$ be the
projection to $\alpha$-th coordinate. Then $p_{\alpha}\circ
f^{\Lambda}(g)=f_{\alpha}\circ g(\alpha),\alpha\in\Lambda$. This shows that
$f^{\Lambda}:\mathbb{N}^{\Lambda}\mapsto\prod_{\alpha\in\Lambda}A_{\alpha}$ is
continuous and onto. It is well known that $\mathbb{N}^{\Lambda}$ is separable
and, hence almost separable. Therefore $\prod_{\alpha\in\Lambda}A_{\alpha}$ is
almost separable. Since $\prod_{\alpha\in\Lambda}A_{\alpha}$ is almost dense
in $\prod_{\alpha\in\Lambda}X_{\alpha}$ , $\prod_{\alpha\in\Lambda}X_{\alpha}$
is almost separable. ∎

######  Definition 3.6.

A topological space $X$ is called functionally Hausdorff if for any two
distinct points $a,b\in X$ there exists $f\in C(X)$ such that $f(a)=0$ and
$f(b)=1$.

######  Lemma 3.7.

If $X$ a functionally Hausdorff space, then for any two distinct points
$a,b\in X$ there exist two distinct cozero sets $C,D$ in $X$ such that $a\in
C$ and $b\in D$.

###### Proof.

$X$ is functionally Hausdorff. Then for $a,b\in X$ with $a\neq b$ there exists
$f\in C(X)$ such that $f(a)=0$ and $f(b)=1$. Let $C=\\{x\in
X:f(x)<\frac{1}{2}\\}=((f-\underline{\frac{1}{2}})\wedge\underline{0})^{-1}(\mathbb{R}\setminus\\{0\\})$
and $D=\\{x\in
X:f(x)>\frac{1}{2}\\}=((f-\underline{\frac{1}{2}})\vee\underline{0})^{-1}(\mathbb{R}\setminus\\{0\\})$,
where $(g\vee h)(x)=\max\\{g(x),h(x)\\}$ and $(g\wedge
h)(x)=\min\\{g(x),h(x)\\}$ for all $x\in X$. Thus $C,D$ are disjoint cozero
sets in $X$ and $a\in C,b\in D$. This completes the proof. ∎

######  Theorem 3.8.

Let $X=\prod_{\alpha\in\Lambda}X_{\alpha}$ be almost separable space, where
each $X_{\alpha}$ is functionally Hausdorff and contains at least two points.
Then each $X_{\alpha}$ is almost separable and $card(\Lambda)\leq c$.

###### Proof.

Let $D$ be a countable ,almost dense subset of $X$. Consider the projection
function $p_{\alpha}:X\mapsto X_{\alpha}$ to the $\alpha$-th coordinate. Then
the function $p_{\alpha}:X\mapsto X_{\alpha}$ is continuous. Since continuous
image of an almost separable space is almost separable, therefore each
$X_{\alpha}$ is almost separable.

For every $\alpha\in\Lambda$, let $a_{\alpha},b_{\alpha}\in X_{\alpha}$ with
$a_{\alpha}\neq b_{\alpha}$. Since each $X_{\alpha}$ is functionally
Hausdorff, therefore there exist disjoint cozero sets $C_{\alpha},D_{\alpha}$
in $X_{\alpha}$ such that $a_{\alpha}\in C_{\alpha}$ and $b_{\alpha}\in
D_{\alpha}$. $p_{\alpha}^{-1}(C_{\alpha})$ is a non-empty cozero set in $X$.
By Theorem 2.15 $K_{\alpha}=D\cap p_{\alpha}^{-1}(C_{\alpha})\neq\emptyset$.

Define a function $\phi:\Lambda\mapsto\mathcal{P}(D)$ by
$\phi(\alpha)=K_{\alpha}$. Now $p_{\alpha}^{-1}(C_{\alpha})\cap
p_{\alpha}^{-1}(D_{\beta})$ is a non-empty cozero set in $X$. Then there exist
$x\in D\cap p_{\alpha}^{-1}(C_{\alpha})\cap p_{\alpha}^{-1}(D_{\beta})$ by
Theorem 2.15. So, $x\in K_{\alpha}$ and $x\notin K_{\beta}$. Therefore
$K_{\alpha}\neq K_{\beta}$. Thus $\phi$ is injective. Hence $card(\Lambda)\leq
card(\mathcal{P}(D))=c$. This completes the proof. ∎

######  Theorem 3.9.

For an almost separable space $X$, the cardinality of $C(X)$ is less than or
equal to $c$.

###### Proof.

Let $A$ be countable almost dense subset of $X$. Define a map
$\phi:C(X)\mapsto C(A)$ by $\phi(f)=f\lvert_{A}$. We sow that $\phi$ is
injective mapping. Let $\phi(f)=\phi(g)$, where $f,g\in C(X)$. Then
$f\lvert_{A}=g\lvert_{A}$. Let $h=f-g$. Then $h\in C(X)$ and $h(A)=\\{0\\}$.
Since $A$ is almost dense in $X$, then $h(X)=\\{0\\}$. Therefore $\phi$ is
injective. Since the cardinality of $C(A)$ is less that or equal to $c$,
therefore the cardinality of $C(X)$ is less than or equal to $c$. ∎

######  Corollary 3.10.

If an almost separable space has a uncountable closed discrete subspace, then
it is not normal.

######  Theorem 3.11.

Let $X$ be functionally Hausdorff, almost separable space. Then the
cardinality of $X$ is atmost $2^{c}$.

###### Proof.

Consider the function $\psi:X\mapsto\mathcal{P}(C(X))$ by $\psi(x)=\\{f\in
C(X):f(x)=0\\}$. Use the functionally Hausdorff property of $X$ to conclude
that $\psi$ is injective. By the Theorem 3.9, cardinality of $C(X)$ is $c$.
Thus cardinality of $X$ is atmost $2^{c}$.

Alternative proof of the above theorem: Let $X$ be a functionally Hausdorff,
almost separable space. Let $\tau_{w}$ be the weak topology on $X$ induced by
$C(X)$. Then $(X,\tau_{w})$ is completely regular Hausdorff space show that
almost dense set becomes dense. Thus $(X,\tau_{w})$ is a separable Hausdorff
space. It is known that the cardinality of a separable Hausdorff space is
atmost $2^{c}$. ∎

######  Theorem 3.12.

Let $Y$ be almost dense in $X$ and $Y$ be almost separable as a subspace. Then
$X$ is almost separable.

###### Proof.

Let $A$ be countable almost dense subset of $Y$. Let $f\in C(X)$ such that
$f(A)=\\{0\\}$. Then $f\lvert_{Y}\in C(Y)$ and $f\lvert_{Y}(A)=\\{0\\}$. This
implies that $f\lvert_{Y}(Y)=\\{0\\}$ as $A$ is almost dense in $Y$. Since $Y$
is almost dense in $X$, then $f(X)=\\{0\\}$. Therefore $A$ is countable almost
dense subset of $X$. Hence $X$ is almost separable space. ∎

##  4\. Baire Category Like Theorem

######  Theorem 4.1.

For a topological space $X$ the following are equivalent:
$(i)X$ is pseudocompact
$(ii)$ If $\\{F_{n}:n\in\mathbb{N}\\}$ is a sequence of zero sets of $X$with
finite intersection property , Then
$\bigcap_{n=1}^{\infty}F_{n}\neq\emptyset$.
$(iii)$ If $\\{U_{n}:n\in\mathbb{N}\\}$ is a countable cover of $X$ consisting
of cozero sets , there exists a finite subcover.

######  Theorem 4.2.

Let $X$ be a Hausdorff space. Given a non-empty cozero-set $U$ and $x\in U$
there exists a cozero-set $V$ and a zero-set $F$ such that $x\in V\subset
F\subset U$.

###### Proof.

Let $f:X\longrightarrow[0,1]$ such that $U=f^{-1}(0,1]=X-f^{-1}(\\{0\\}).$
Since $x\in U,f(x)>0$. Choose $\delta>0$ such that $0<f(x)-\delta<f(x)$. Let
$V=f^{-1}(f(x)-\delta,1]),F=f^{-1}[f(x)-\delta,1]$. Then $x\in V\subset\subset
F\subset U$. Now $(f(x)-\delta,1]\subset[0,1]$ is an open set and hence a
cozero-set and $[f(x)-\delta,1]$ is a closed subset of $[0,1]$ and hence a
zero-set. Then $V=f^{-1}(f(x)-\delta,1]$ is a cozero-set and
$F=f^{-1}[f(x)-\delta,1]$ is a zero-set of $X$. ∎

######  Theorem 4.3.

(Baire Category Like Theorem) Let $X$ be a Hausdorff, pseudocompact space. If
$\\{U_{n}:n\in\mathbb{N}\\}$ is a sequence of almost dense cozero-sets of $X$,
then $\bigcap_{n=1}^{\infty}U_{n}$ is a non-empty almost dense subset of $X$.

###### Proof.

Write $D=\bigcap_{n=1}^{\infty}U_{n}$. To show $D\neq\emptyset$ and $D$
intersects every non-empty cozero-set. Let $V$ be a non-empty cozero-set and
let $x\in V$. Since $U_{1}$ is almost dense , $V\cap U_{1}\neq\emptyset$ and
is a cozero-set . Let $x_{1}\in V\cap U_{1}$. By theorem $(4.2)$, $\exists$
cozero-set $V_{1}$ and a zero-set $F_{1}$ such that $x_{1}\in V_{1}\subset
F_{1}\subset V\cap U_{1}$. Now $V_{1}\neq\emptyset$ cozero-set
$\Longrightarrow V_{1}\cap U_{2}\neq\emptyset$ and is a zero-set. Let
$x_{2}\in V_{1}\cap U_{2}.\exists V_{2}$ a non empty co-zero set and $F_{2}$ ,
a zero-set , such that $x_{2}\in V_{2}\subset F_{2}\subset V_{1}\cap
U_{2}\subset V\cap U_{1}\cap U_{2}$. Now $V_{2}\neq\emptyset$ cozero-set,
$V_{2}\cap U_{3}\neq\emptyset$ and is a cozero -set. Let $x_{3}\in V_{2}\cap
U_{3}$. $\exists V_{3}$, a cozero-set, and $F_{3}$ , a zero set , such that
$x_{3}\in V_{3}\subset F_{3}\subset V_{2}\cap U_{3}\subset V\cap U_{1}\cap
U_{2}\cap U_{3}$. Proceeding in this way we obtain non-empty cozero-set
$V_{n+1}$, zero-set $F_{n+1}$ such that $x_{n+1}\in V_{n+1}\subset
F_{n+1}\subset V_{n}\cap U_{n+1}\subset V\cap U_{1}\cap U_{2}\cap
U_{3}\cap......\cap U_{n+1}$ for all $n\geq 0$. Note that $F_{n+1}\subset
F_{n}$ and $F_{n}^{\prime}$ s are non-empty zero-sets. Since $X$ is
pseudocompact , in virtue of the theorem $(4.1)$,
$\bigcap_{n=1}^{\infty}F_{n}\neq\emptyset$. Hence
$\bigcap_{n=1}^{\infty}F_{n}\subset\bigcap_{n=1}^{\infty}(V\cap U_{1}\cap
U_{2}\cap U_{3}\cap.......\cap U_{n})=V\bigcap_{n=1}^{\infty}U_{n}$ so that
$V\cap(\bigcap_{n=1}^{\infty}U_{n})\neq\emptyset$. i.e, $V\cap
D\neq\emptyset$. Since $V$ is an arbitary non-empty cozero-set , $D$ is almost
dense. ∎

We know that separability is not a hereditary property. Niemytzky’s plane is a
well known example. Same is true about almost separability as well.
Niemytzky’s plane provides an example in this case also.

We now finish our paper giving relations among the different types of
separability notions which are defined earlier.

Strong sequentially separable $\Rightarrow$ Sequentially separable
$\Rightarrow$ Separable $\Rightarrow$ Almost separable.

Acknowledgments: The authors would like to thank to Professor Alan Dow for
Theorem 2.4. Also the authors wish to thank Professor Asit Baran Raha for
their valuable suggestions that improved the article.

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