On the class of weakly almost contra-T*-continuous functions

TL;DR

This paper introduces and explores weakly almost contra-$T^*$-continuous functions, a new class of functions in operator topological spaces, providing characterizations, propositions, and counterexamples to understand their properties.

Contribution

It defines the new class of weakly almost contra-$T^*$-continuous functions and investigates their fundamental properties and characterizations.

Findings

New class of functions introduced

Several basic properties proved

Counterexamples provided

Abstract

The aim of this paper is to introduce and investigate a new class of functions called weakly almost contra-$T^*$-continuity which is defined as a function from an operator topological space $(X, \tau, T)$ into an arbitrary topological space $(Y, \delta)$. Furthermore, some new characterizations, several basic propositions are proved and some relevant counterexamples are provided.

Equations

$$\mathrm{pCl}(S)=S\cup \mathrm{Cl}(\mathrm{Int}(S)),$$

$$U\subseteq \mathrm{Int}(\mathrm{Cl}(U))=T(U).$$

$$\mathrm{pCl}(S)=S\cup \mathrm{Cl}(\mathrm{Int}(S)).$$

$$X={\cup }_{i=1}^n(f^{-1}(V_{y_i})),$$

$$Y=f(X)=f({\cup }_{i=1}^n(f^{-1}(V_{y_i})))={\cup }_{i=1}^n(V_{y_i}),$$

$$T^{\ast }\mathrm{Cl}(f^{-1}(\mathrm{Cl}(W)))\subseteq f^{-1}(Y\setminus \mathrm{Cl}(V)).$$

$$(U\times \mathrm{Int}(\mathrm{Cl}(V)))\cap G(f),$$

$$T^{\ast }\mathrm{Cl}(f^{-1}(\mathrm{Cl}(V)))\subseteq f^{-1}(Y\setminus \mathrm{Cl}(W)).$$

$$(x,y)\in (X\setminus T^{\ast }\mathrm{Cl}(f^{-1}(\mathrm{Cl}(V))))\subseteq X\times Y\setminus G(f)$$

Full text

1. Introduction

In the literature, a number of generalizations of open sets and its continuous functions have been considered. Indeed, many mathematicians worked in this area and made great contributions to develop several types of almost contra-continuous and weakly almost contra-continuous functions. These functions which are defined between two an arbitrary topological spaces have been discussed extensively in the literature. For general reference, we refer the reader to J. Dontchev [7] in 1996, J. Dontchev and T. Noiri [8] in 1999, M. Caldas and S. Jafari, [6] in 2001 and E. Ekici [9] in 2004. C. W. Baker studied and developed several types of weakly contra-continuous functions (see for instance [4], [5]). Moreover, many of the related concepts studied well such that this subject has been received much attention in the last decade. Among others, see [9], [10], [11], [12] and [13].

H. J. Mustafa et al. used a different technique to define the continuity of functions from an operator topological space $(X,\tau,T)$, being a topological space with an operator $T$ associated with the topology $\tau$, into an arbitrary topological space $(Y,\delta)$, we refer the reader to [20] [21]. Using the concept of $T^{*}$-open set in [21] they introduced and studied almost contra-$T^{*}$-continuous functions, several properties and characterizations of these functions are considered. In this paper, we continue this line to explore a new approach to weakly almost contra-continuity such that our goal is to introduce some definitions and investigate various properties of a new category of functions called weakly almost contra-$T^{*}$-continuous in topological spaces via utilizing the concept of $T^{*}$-open set.

In the sequel, we will present a number of concepts which are linked to our investigations. First, in Section 2 we give the basic definitions and notations. Afterward, in Section 3, we will pay our attention to discuss weakly almost contra-$T^{*}$-continuous functions and its relationships to several other close concepts. The following are the main results of this paper:

• (1)

Let f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) be a function where $T(S)=\mathrm{Int(Cl}(S))$. Then $f$ is weakly almost contra-$T^{*}$-continuous if and only if, whenever $S$ is regular closed in $Y$, $V$ is regular open subset of $Y$, and $S\subseteq V$, then $\mathrm{Cl(Int}(f^{-1}(S)))\subseteq f^{-1}(V)$.

• (2)

If f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is an almost-$T^{*}$-continuous function, then $f$ is weakly almost contra-$T^{*}$-continuous.

• (3)

If f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is a weakly almost contra-$T^{*}$-continuous function, where $(Y,\delta)$ is an extremely disconnected space, then $f$ is almost-$T^{*}$-continuous.

• (4)

Let $f$ be a function from an operator topological space $(X,\tau,T)$ into an extremely disconnected space $(Y,\delta)$. Then the weakly almost contra-$T^{*}$-continuity is equivalent to the almost-$T^{*}$-continuity.

• (5)

If f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is an almost contra-$T^{*}$-continuous function, then $f$ is weakly almost contra-$T^{*}$-continuous.

• (6)

Suppose that f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is a weakly almost contra-$T^{*}$-continuous function, then $f$ is slightly contra-$T^{*}$-continuous.

• (7)

Let f:(X,\tau,T)\,\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\,(Y,\delta) is a weakly almost contra-$T^{*}$-continuous surjection and let $Y$ be a $\Sigma$-space. If $X$ is contra-$T^{*}$-compact then $Y$ is $R$-compact.

• (8)

If the function f:(X,\tau,T)\,\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\,(Y,\delta) is weakly almost contra-$T^{*}$-continuous and $Y$ is Urysohn, then $G(f)$ has a $T^{*}$-regular and contra $T^{*}$-regular graph as well.

• (9)

Let f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) be weakly almost contra-$T^{*}$-continuous and the images of g$T^{*}$r-closed sets are regular closed, then $f$ is a$T^{*}$r-irresolute.

• (10)

If f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is almost g$T^{*}$r-continuous and a$T^{*}$r-irresolute, then $f$ is weakly almost contra-$T^{*}$-continuous.

2. Preliminaries

In this section, we will present the definitions and the basic concepts that play an important role in this paper. The closure of $S$ will be denoted by $\mathrm{Cl}(S)$. The interior of $S$ will be denoted by $\mathrm{Int}(S)$.

Definition 1**.**

A subset $S$ of a topological space $(X,\tau)$ is said to be:

• $\bullet$

regular open if $S=\mathrm{Int(Cl}(S))$, regular closed if $S=\mathrm{Cl(Int}(S))$ [27].

• $\bullet$

pre-open if $S\subseteq\mathrm{Int(Cl}(S))$, the complement of pre-open is a pre-closed [18].

• $\bullet$

semi-open if $S\subseteq\mathrm{Cl(Int}(S))$, the complement of semi-open is a semi-closed [17].

• $\bullet$

$\alpha$-open if $S\subseteq\mathrm{Int(Cl(Int}(S)))$, the complement of $\alpha$-open is a $\alpha$-closed [22].

• $\bullet$

$\beta$-open if $S\subseteq\mathrm{Cl(Int(Cl}(S)))$, the complement of $\beta$-open is a $\beta$-closed [1].

The $\beta$-closure of a set $S$ denoted by $\beta\mathrm{Cl}(S)$, is the intersection of all $\beta$-closed sets containing $S$. The $\beta$-interior of a set $S$ denoted by $\beta\mathrm{Int}(S)$, is the union of all $\beta$-open sets contained in $S$. The preclosure, preinterior, semiclosure and semiinterior of a set $S$ denoted by $\mathrm{pCl}(S),\mathrm{pInt}(S),\mathrm{sCl}(S)\ \text{and}\ \mathrm{sInt}(S)$, respectively, are defined analogously. We say that $V$ is clopen subset if $V$ is both open and closed. Furthermore, we have for any set $S$ that

[TABLE]

for more details see [9].

Definition 2**.**

A function f:(X,\tau)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is said to be continuous (pre-continuous [18], semi-continuous [17], $\alpha$-continuous [24], $\beta$-continuous [1], resp.) if $f^{-1}(V)$ is open (pre-open, semi-open, $\alpha$-open, $\beta$-open, resp.) in $X$ for each open subset $V\subseteq Y$.

Definition 3**.**

A function f:(X,\tau)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is said to be contra-continuous [7] (contra-pre-continuous [15], contra-semi-continuous [8], contra-$\alpha$-continuous [16], contra-$\beta$-continuous [6], resp.) if $f^{-1}(V)$ is closed (pre-closed, semi-closed, $\alpha$-closed, $\beta$-closed, resp.) in $X$ for each open subset $V\subseteq Y$.

Definition 4**.**

A function f:(X,\tau)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is said to be almost contra-continuous [25](almost contra-pre-continuous [9], almost contra-semi-continuous [14], almost contra-$\alpha$-continuous [23], almost contra-$\beta$-continuous [6], resp.) if $f^{-1}(V)$ is closed (pre-closed, semi-closed, $\alpha$-closed, $\beta$-closed, resp.) in $X$ for every regular open $V$ of $Y$.

Definition 5**.**

A function f:(X,\tau)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is said to be weakly contra-continuous [4] (weakly contra-pre-continuous [4], weakly contra-$\beta$-continuous[5], resp.) provided that, whenever $S\subseteq V\subseteq Y$, $S$ is closed in $Y$, and $V$ is open in $Y$, then $\mathrm{Cl}f^{-1}(S)\subseteq f^{-1}(V)$ ($\mathrm{pCl}f^{-1}(S)\subseteq f^{-1}(V)$, $\mathrm{\beta Cl}f^{-1}(S)\subseteq f^{-1}(V)$, resp.) in $X$.

Definition 6**.**

[20] Let $(X,\tau)$ be a topological space and $P(X)$ be the power set of $X$. A function T:P(X)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>P(X) is said to be an operator associated with topology $\tau$ on $X$ if $U\subseteq T(U)$ for all $U\in\tau$ and the triple $(X,\tau,T)$ is called an operator topological space.

Example 1*.*

• $\bullet$

If $T$ is the identity operator, i.e., $T(S)=S$, then the triple $(X,\tau,T)$ will reduces to $(X,\tau)$, thus the operator topological space is the ordinary topological space.

• $\bullet$

Let $(X,\tau)$ be any topological space and function T:P(X)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>P(X) such that $T(S):=\mathrm{Int(Cl}(S))$ for any $S\subseteq X$. Notice that if $U$ is open in $X$, then

[TABLE]

Consequently, $T$ is an operator associated with the topology $\tau$ on $X$ and the triple $(X,\tau,T)$ is an operator topological space.

Definition 7**.**

[21] Let $(X,\tau,T)$ be an operator topological space and $S\subseteq X$, then $S$ is said to be $T^{*}$-open if $S\subseteq T(S)$ (observe that $S$ not necessarily open). The complement of $T^{*}$-open is called $T^{*}$-closed.

Remark 1*.*

• $\bullet$

$T^{*}\mathrm{Cl}(S):=\cap\ \{U\mid U\ \text{is}\ T^{*}\text{-closed},\ U\supseteq S\}$,

• $\bullet$

if $T(S)=\mathrm{Int(Cl}(S))$, where $S\subseteq X$ then $T^{*}$-open set is exactly the pre-open set and $T^{*}\mathrm{Cl}(S)\equiv\mathrm{pCl}(S)$,

• $\bullet$

if $T(S)=\mathrm{Cl(Int}(S))$, where $S\subseteq X$ then $T^{*}$-open set is exactly the semi-open set and $T^{*}\mathrm{Cl}(S)\equiv\mathrm{sInt}(S)$.

Definition 8**.**

A function f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is said to be [21]:

• •

$T^{*}$-continuous if $f^{-1}(V)$ is $T^{*}$-open in $X$ for each open subset $V\in\delta$.

• •

almost $T^{*}$-continuous if $f^{-1}(V)$ is $T^{*}$-open in $X$ for every regular open subset $V$ of $Y$.

• •

contra-$T^{*}$-continuous if $f^{-1}(V)$ is $T^{*}$-closed in $X$ for every open subset $V$ of $Y$.

• •

almost contra-$T^{*}$-continuous if $f^{-1}(V)$ is $T^{*}$-closed in $X$ for every regular open subset $V$ of $Y$.

Definition 9**.**

A function f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is called:

• •

slightly contra-$T^{*}$-continuous if, for every $x\in X$ and every clopen subset $V$ of $Y$ containing $f(x)$ there exists a $T^{*}$-closed subset $f^{-1}(V)$ of $X$ with $x\in f^{-1}(V)$ and $f(f^{-1}(V))\subseteqq V$.

• •

weakly contra-$T^{*}$-continuous if for any $S\subseteq V\subseteq Y$, $S$ closed, $V$ open in $Y$, we have $T^{*}$Cl$f^{-1}(S)\subseteq f^{-1}(V)$.

• •

weakly almost contra-$T^{*}$-continuous if for every regular open subset $V$ of $Y$ and every regular closed subset $S$ of $Y$ with $S\subseteq V$, we have $T^{*}$Cl$f^{-1}(S)\subseteq f^{-1}(V)$ in $X$.

Some examples of weakly almost contra-$T^{*}$-continuous functions will be shown later.

Definition 10**.**

Let $(X,\tau)$ be a topological space, then $X$ is said to be extremely disconnected [2] whenever the closures of open sets are open.

3. Weakly almost contra-$T^{*}$-continuous functions

Lemma 1**.**

Let f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) be a function where $T(S)=\mathrm{Int(Cl}(S))$. Then $f$ is weakly almost contra-$T^{*}$-continuous if and only if, whenever $S$ is regular closed in $Y$, $V$ is regular open subset of $Y$, and $S\subseteq V$, then $\mathrm{Cl(Int}(f^{-1}(S)))\subseteq f^{-1}(V)$.

Proof.

Since $T(S)=\mathrm{Int(Cl}(S))$, $T^{*}$-openness plays the same role as pre-openness and $T^{*}\mathrm{Cl}(S)\equiv\mathrm{pCl}(S)$. Turning on to our proof, we know that

[TABLE]

But pCl($f^{-1}(S))\subseteq f^{-1}(V)$, then $f^{-1}(S)\cup\mathrm{Int(Cl(}f^{-1}(S)))\subseteq f^{-1}(V)$, therefore $\mathrm{Int(Cl(}f^{-1}(S)))\subseteq f^{-1}(V)$. ∎

Remark 2*.*

• •

Every contra-$T^{*}$-continuous function are automatically weakly contra-$T^{*}$-continuous function, since $S\subseteq V$ implies $f^{-1}(S)\subseteq f^{-1}(V)$ and T^{*}$$f^{-1}(S)\subseteq T^{*}$$f^{-1}(V). If $f^{-1}(V)$ is $T^{*}$-closed, then T^{*}$$f^{-1}(V)\subseteq f^{-1}(V), so we conclude that T^{*}$$f^{-1}(S)\subseteq f^{-1}(V).

• •

Following the same technique as above one can check that every weakly contra-$T^{*}$-continuous function is a weakly almost contra-$T^{*}$-continuous function.

Proposition 2**.**

If f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is an almost-$T^{*}$-continuous function, then $f$ is weakly almost contra-$T^{*}$-continuous.

Proof.

Suppose that $f$ is an almost-$T^{*}$-continuous function. First, fix that $S\subseteq V\subseteq Y$ such that $S$ is regular closed in $Y$ and $V$ is regular open in $Y$. Now, $f^{-1}(S)$ is $T^{*}$-closed, under the hypothesis that $f$ is an almost-$T^{*}$-continuous function and thus, $T^{*}$Cl$f^{-1}(S)\subseteq f^{-1}(S)\subseteq f^{-1}(V).$ Consequently, $f$ is a weakly almost contra-$T^{*}$-continuous function. ∎

Proposition 3**.**

If f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is a weakly almost contra-$T^{*}$-continuous function, where $(Y,\delta)$ is an extremely disconnected space, then $f$ is almost-$T^{*}$-continuous.

Proof.

Let $V$ be a regular closed subset of $Y$. Under the conditions stated above that $(Y,\delta)$ is extremely disconnected, $V$ is clopen and hence $V$ is also regular open. Therefore, $T^{*}$Cl$f^{-1}(V)\subseteq f^{-1}(V)$. By assumption $f$ is weakly almost contra-$T^{*}$-continuous functions, from what we conclude that $f^{-1}(V)$ is $T^{*}$-closed. Thus, $f$ is an almost-$T^{*}$-continuous function. ∎

One can prove immediately the next corollary from Propositions 2 and 3.

Corollary 4**.**

Let $f$ be a function from an operator topological space $(X,\tau,T)$ into an extremely disconnected space $(Y,\delta)$. Then the weakly almost contra-$T^{*}$-continuity is equivalent to the almost-$T^{*}$-continuity.

Proposition 5**.**

Let f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) be an almost contra-$T^{*}$-continuous function, then $f$ is weakly almost contra-$T^{*}$-continuous.

Proof.

Assume that f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is an almost contra-$T^{*}$-continuous function. Let $S$ is a regular closed in $Y$ and $V$ is a regular open in $Y$ such that $S\subseteq V\subseteq Y$, since $f$ satisfies the property of almost contra-$T^{*}$-continuous. Therefore, $f^{-1}(V)$ is $T^{*}$-closed and therefore, $T^{*}$Cl$f^{-1}(S)\subseteq$ $T^{*}$Cl$f^{-1}(V)\subseteq f^{-1}(V).$ We thus obtain $f$ is a weakly almost contra-$T^{*}$-continuous function. ∎

Proposition 6**.**

Suppose that f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is a weakly almost contra-$T^{*}$-continuous function, then $f$ is slightly contra-$T^{*}$-continuous.

Proof.

We consider f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) to be weakly almost contra-$T^{*}$-continuous and $V$ be a regular clopen (i.e., $V$ is regular open and regular closed) subset of $Y$. Then, since $V\subseteq V\subseteq Y$. This implies that $T^{*}$Cl$f^{-1}(V)\subseteq f^{-1}(V)$. Therefore, $f^{-1}(V)$ is $T^{*}$-closed and $f$ is a slightly contra-$T^{*}$-continuous function, as wanted to be shown.∎

Corollary 7**.**

If f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is a weakly contra-$T^{*}$-continuous function, then $f$ is slightly contra-$T^{*}$-continuous.

Consequently, from what we have already proved, one can consider the following diagram: (C.= continuous)

weakly contra-C.

$\Downarrow$

weakly contra-$T^{*}$-C.

$\Downarrow$

almost-$T^{*}$-C. $\Longrightarrow$ weakly almost contra-$T^{*}$-C. $\Longrightarrow$ slightly contra-$T^{*}$-C.

$\Uparrow$

                                           almost contra-$T^{*}$-C.

The next examples show that, in general, none of the above implications are reversible.

Example 2*.*

• $\bullet$

Let f:(\mathbb{R},\tau,T)\,\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\,(\mathbb{R},\delta) be the identity function such that $\tau=\{\mathbb{R},\emptyset,\{0\}\}$ and T:P(\mathbb{R})\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>P(\mathbb{R}) is defined by $T(S)=\mathrm{Int(Cl(}S))$ and $\delta$ is the usual topology on $\mathbb{R}$. Since $(\mathbb{R},\delta)$ is connected, $f$ is slightly contra-$T^{*}$-continuous function. However, $f$ is not weakly almost contra-$T^{*}$-continuous. To check this one can consider $S=[0,1]$ and $V=(-2,2)$, then $S$ is regular closed in $(\mathbb{R},\delta)$ and $V$ is regular open in $(\mathbb{R},\delta)$ with $S\subseteq V$, but $T^{*}$Cl$(f^{-1}(S))=\mathrm{pCl}(f^{-1}(S))\nsubseteq f^{-1}(V).$

• $\bullet$

Let us consider the same identity function from $(X,\tau,T)$ into $(X,\delta)$, assume that $X=\{a,b,c\}$ have the following topologies $\tau=\{X,\emptyset,\{a\}\}$, and $\delta=\{X,\emptyset,\{c\},\{a,c\},\{b,c\}\}$ and T:P(X)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>P(X) defined by $T(S)=\mathrm{Int(Cl}(S))$. Hence, the only regular open sets in $(X,\delta)$ are $X$ and $\emptyset$ where $f$ is weakly almost contra-$T^{*}$-continuous. Moreover, $f$ is not weakly contra-$T^{*}$-continuous. Since for $S=\{a\}$ and $V=\{a,c\}$ in the same space $(X,\delta)$, indeed $T^{*}$Cl$(f^{-1}(S))=\mathrm{pCl}(f^{-1}(S))\nsubseteq f^{-1}(V).$

• $\bullet$

Suppose that f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(X,\delta) is the identity function, where $X=\{a,b,c\}$ and its topology given by $\tau=\{X,\emptyset,\{a\},\{b\},\{a,b\}\}$ such that T:P(X)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>P(X) defined as following $T(S)=\mathrm{Int(Cl}(S))$. Then $f$ is almost $T^{*}$-continuous but not almost contra-$T^{*}$-continuous. Note that $S=\{a\}$ is regular open in $X$ and $f^{-1}(S)$ not $T^{*}$-closed.

• $\bullet$

Let f:(X,\tau,T)\,\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\,(X,\delta) be the identity function, defined by $X=\{a,b,c\}$ where its topologies are described by $\tau=\{X,\emptyset,\{c\}\}$ and $\delta=\{X,\emptyset,\{a\},\{b\},\{a,b\}\}.$ As above, T:P(X)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>P(X) is given by the following form $T(S)=\mathrm{Int(Cl}(S))$. Under the above assumptions, $f$ is an almost contra-$T^{*}$-continuous function. Consequently, $S=\{a\}$ is regular open in $(X,\delta)$, but $f^{-1}(S)$ is not $T^{*}$-open.

Definition 11**.**

Let $(X,\tau)$ be a topological space, then $X$ is said to be:

• $\bullet$

$\Sigma$-space [19] provided that every open set is the union of regular closed sets.

• $\bullet$

$R$-compact [26] if every regular open cover of $X$ has a finite subcover.

Taking into account the operator topological space $(X,\tau,T)$ we define the following:

Definition 12**.**

Let $f:(X,\tau,T)$ be an operator topological space, then $X$ is said to be contra $T^{*}$-compact if every cover of $X$ by $T^{*}$-closed sets has a finite subcover.

Proposition 8**.**

Let f:(X,\tau,T)\,\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\,(Y,\delta) is a weakly almost contra-$T^{*}$-continuous surjection and let $Y$ be a $\Sigma$-space. If $X$ is contra-$T^{*}$-compact then $Y$ is $R$-compact.

Proof.

Let $C$ be a cover of $Y$ by regular open sets. Let $y\in Y$ and let $V_{y}\in C$ such that $y\in V_{y}$. Since $Y$ is a $\Sigma$-space, there exists a regular closed set $S_{y}$ such that $y\in S_{y}\in V_{y}$. Since $f$ is weakly almost contra-$T^{*}$-continuous, $T^{*}$Cl$(f^{−1}(S_{y}))\subseteq f^{-1}(V_{y})$. It follows that {$T^{*}$Cl$(f^{-1}(S_{y}))|y\in Y\}$ is a cover of $X$ by $T^{*}$-closed sets. Since $X$ is contra $T^{*}$-compact, there exists a finite subcover {$T^{*}$Cl$(f^{-1}(V_{y_{i}}))|i=1,2,...,n\}$ It then follows that

[TABLE]

[TABLE]

this shows that $Y$ is $R$-compact.

∎

Definition 13**.**

A topological space $(X,\tau)$ is said to be Urysohn [3], if for every pair of distinct points $x$ and $y$ in $X$, there exist open sets $U$ and $V$ such that $x\in U,\ y\in V$ and $\mathrm{Cl}(U)\cap\mathrm{Cl}(V)=\emptyset$.

Definition 14**.**

Let f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) be given.

• $\bullet$

The graph $G(f)$ of a function $f$ is said to be $T^{*}$-regular whenever $(x,y)\in X\times Y\setminus G(f)$ there exist a $T^{*}$-closed set $U$ in $X$ containing $x$ and a regular open set $V$ in $Y$ containing $y$ such that $(U\times V)\cap G(f)=\emptyset$. This is equivalent to $f(U)\cap V=\emptyset.$

• $\bullet$

$f$ has a contra $T^{*}$-regular graph under the condition that for every $(x,y)\in X\times Y\setminus G(f)$ there exist a $T^{*}$-closed set $U$ in $X$ containing $x$ and a regular closed set $V$ in $Y$ containing $y$ such that $(U\times V)\cap G(f)=\emptyset$.

• $\bullet$

For any an operator topological space $(X,\tau,T)$ and $S\subseteq X$, we call $S$ generalized $T^{*}$-regular closed (briefly g$T^{*}$r-closed) if it is satisfying $T^{*}\mathrm{Cl}(S)\subseteq U$ whenever $S\subseteq U$ and $U$ is regular open.

• $\bullet$

$f$ is called approximately $T^{*}$-regular irresolute (briefly a$T^{*}$r-irresolute) if $T^{*}\mathrm{Cl}(S)\subseteq f^{−1}(V)$ whenever $V$ is regular open, $S$ is g$T^{*}$r-closed, and $S\subseteq f^{−1}(V).$

• $\bullet$

$f$ is said to be an almost g$T^{*}$r-continuous function whenever $f^{-1}(S)$ is $gT^{*}r$-closed for every regular closed subset $S$ of $Y$.

Proposition 9**.**

If the function f:(X,\tau,T)\,\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\,(Y,\delta) is weakly almost contra-$T^{*}$-continuous and $Y$ is Urysohn, then $G(f)$ has a $T^{*}$-regular and contra $T^{*}$-regular graph as well.

Proof.

Let $(x,y)\in X\times Y\setminus G(f)$. Then, since $y\neq f(x)$ and $Y$ is Urysohn, there exist open sets $V$ and $W$ in $Y$ such that $y\in V$ and $f(x)\in W$ and $\mathrm{Cl}(V)\cap\mathrm{Cl}(W)=\emptyset$. Then we see that $\mathrm{Cl}(W)\subseteq Y\setminus\mathrm{Cl}(V)$,  $\mathrm{Cl}(W)$ is regular closed, and $Y\setminus\mathrm{Cl}(V)$ is regular open. Since $f$ is weakly almost contra-$T^{*}$-continuous,

[TABLE]

It then follows that $(x,y)\in T^{*}\mathrm{Cl}(f^{-1}(\mathrm{Cl}(W)))\times\mathrm{Int(Cl}(V))$.

Let $U=T^{*}\mathrm{Cl}(f^{-1}(\mathrm{Cl}(W)))$, $U$ is $T^{*}$-closed. Since $\mathrm{Int(Cl}(V))$ is regular open,

[TABLE]

which proves that $G(f)$ is $T^{*}$-regular.

To prove the second property, let $(x,y)\in X\times Y\setminus G(f)$. As an above $y\neq f(x)$ and $Y$ is Urysohn, there exist open sets $V$ and $W$ in $Y$ such that $y\in V$ and $f(x)\in W$ and $\mathrm{Cl}(V)\cap\mathrm{Cl}(W)=\emptyset$. Therefore $\mathrm{Cl}(V)\subseteq Y\setminus\mathrm{Cl}(W)$, since $\mathrm{Cl}(W)$ is regular closed, and $Y\setminus\mathrm{Cl}(W)$ is regular open. Moreover, $f$ is weakly almost contra-$T^{*}$-continuous, we obtain

[TABLE]

It then follows that

[TABLE]

and we have that $G(f)$ is a contra $T^{*}$-regular graph.

∎

Proposition 10**.**

Let f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) be weakly almost contra-$T^{*}$-continuous and the images of g$T^{*}$r-closed sets are regular closed, then $f$ is a$T^{*}$r-irresolute.

Proof.

Let $V$ be a regular open subset of $Y$ and let $S$ be a g$T^{*}$r-closed subset of $X$ such that $S\subseteq f^{-1}(V)$. Then $f(S)$ is regular closed and $f(S)\subseteq V$. Since $f$ is weakly almost contra-$T^{*}$-continuous, $T^{*}$Cl($f^{-1}(f(S)))\subseteq f^{-1}(V)$. Therefore $T^{*}$Cl(S)$\subseteq f^{-1}(V)$ and hence $f$ is $aT^{*}r$-irresolute. ∎

Proposition 11**.**

If f:(X,\tau,T)\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\scriptstyle{}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(Y,\delta) is almost g$T^{*}$r-continuous and a$T^{*}$r-irresolute, then $f$ is weakly almost contra-$T^{*}$-continuous.

Proof.

Assume $S\subseteq V\subseteq Y$, where $S$ is regular closed in $Y$ and $V$ is regular open in $Y$. Since $f$ is almost g$T^{*}$r-continuous, $f^{-1}(S)$ is g$T^{*}$r-closed. Then, since $f^{-1}(S)\subseteq f^{-1}(V)$ and $f$ is a$T^{*}$r-irresolute, $T^{*}$Cl($f−1(f(A)))\subseteq f^{-1}(V)$, which proves that $f$ is weakly almost contra-$T^{*}$-continuous. ∎

Acknowledgement

I offer my sincerest gratitude to my supervisor Dr. László Kozma, for carefully reviewing the work, providing useful suggestions.