Graded fuzzy topological spaces

Abstract

In this paper, graded fuzzy topological spaces based on the notion of neighbourhood system of graded fuzzy neighbourhoods at ordinary points are introduced and studied. These graded fuzzy neighbourhoods at ordinary points and usual subsets played the main role in this study.

Keywords:

• neighbourhood systems
• fuzzy filters
• fuzzy neighbourhood filters
• fuzzy topological spaces
• separation axioms

AMS Subject Classifications:

• 54A40

Public Interest Statement

Separation axioms depend on the concept of neighbourhoods and so, for the fuzzy case, fuzzy neighbourhoods or valued fuzzy neighbourhoods means neighbourhoods with some degree in [0, 1] . These grades to be a fuzzy neighbourhood forced the fuzzy separation axioms to be graded. In the fuzzy case, separation axioms are not sharp concepts. For example, there is no $T_0$ topological space, but there are $(\mathit{\alpha },\mathit{\beta })-T_0$ topological spaces depending on the existence of the fuzzy neighbourhood with grade $\mathit{\alpha }$ at a point or the existence of the fuzzy neighbourhood with grade $\mathit{\beta }$ at the other distinct point. In this paper, I introduced these graded fuzzy separation axioms. The main section was for defining a fuzzy neighbourhood system of fuzzy graded neighbourhoods at ordinary points and usual subsets.

1. Introduction

Kubiak (1985) and Sǒstak (1985) introduced the fundamental concept of a fuzzy topological structure as an extension of both crisp topology and fuzzy topology Chang (1968), in the sense that both objects and axioms are fuzzified and we may say they began the graded fuzzy topology. Bayoumi and Ibedou (2001,2002,2002b,2004) introduced and studied the separation axioms in the fuzzy case in Chang’s topology (1968) using the notion of fuzzy filter defined by Gähler (1995a,1995b).

Now, we will try to investigate fuzzy topological spaces in sense of S$\check{o}$stak, not using fuzzy filters but starting from a neighbourhood system of graded fuzzy neighbourhoods at ordinary points and usual sets. From that neighbourhood system, we can build a fuzzy topology in sense of S$\check{o}$stak and moreover, this fuzzy topology is itself the fuzzy topology in sense of Chang associated with the fuzzy neighbourhoterest Satementod filter ${\mathcal{N}}_x$ (Gähler, 1995b) at ordinary point $x\in X$ defined by Gähler. Interior operator and closure operator are defined using these graded fuzzy neighbourhoods; also their associated fuzzy topologies coincide with this fuzzy topology in sense of Chang associated with the fuzzy neighbourhood filter of Gähler. Fuzzy continuous, fuzzy open and fuzzy closed mappings are defined with grades according to these graded fuzzy neighbourhoods.

Separation axioms in the fuzzy case are introduced based on these graded fuzzy neighbourhoods and thus, axioms are graded. These axioms satisfy common results and implications. These graded axioms are a good extension in sense of Lowen (1978). In Fuzzy neuro systems for machine learning for large data sets (2009) and DCPE Co-Training for Classification (2012), there are some applications based on fuzzy sets.

2. Preliminaries

Throughout the paper, let $I_0=(0,1]$ and $I_1=[0,1)$.

A fuzzy topology $\mathit{\tau }:I^X\to I$ is defined by Kubiak (1985) and Sǒstak (1985):

(1)	$\mathit{\tau }(\overline{0})=\mathit{\tau }(\overline{1})=1$,
(2)	$\mathit{\tau }(f\wedge g)\ge \mathit{\tau }(f)\wedge \mathit{\tau }(g)$ for all $f,g\in I^X$,
(3)	$\mathit{\tau }(\underset{j\in J}{\bigvee }{\mathit{\mu }}_j)\ge \underset{j\in J}{\bigwedge }\mathit{\tau }({\mathit{\mu }}_j)$ for any family of ${({\mathit{\mu }}_j)}_{j\in J}\in I^X$.

Let ${\mathit{\tau }}_1$ and ${\mathit{\tau }}_2$ be fuzzy topologies on X. Then, ${\mathit{\tau }}_1$ is finer than ${\mathit{\tau }}_2$ (${\mathit{\tau }}_2$, which is coarser than ${\mathit{\tau }}_1$), denoted by ${\mathit{\tau }}_1\le {\mathit{\tau }}_2$, if ${\mathit{\tau }}_2(\mathit{\mu })\le {\mathit{\tau }}_1(\mathit{\mu })$ for all $\mathit{\mu }\in I^X$.

For each fuzzy set $f\in I^X$, the weak $\mathit{\alpha }$ cut-off f is given by $w_{\mathit{\alpha }}f=\{x\in X\mid f(x)\ge \mathit{\alpha }\}$; the strong $\mathit{\alpha }$ cut-off f is the subset of X, $s_{\mathit{\alpha }}f=\{x\in X\mid f(x)>\mathit{\alpha }\}$.

If T is an ordinary topology on X, then the induced fuzzy topology on X is given by $\mathit{\omega }(T)=\{f\in I^X\mid s_{\mathit{\alpha }}f\in T\text{for all}\mathit{\alpha }\in I_1\}$.

fuzzy filters. Let X be a non- empty set. A fuzzy filter on X (Eklund, 1992; Gähler, 1995a) is a mapping $\mathcal{M}:I^X\to I$ such that the following conditions are fulfilled:

(F1)	$\mathcal{M}(\overline{\mathit{\alpha }})\le \mathit{\alpha }$ holds for all $\mathit{\alpha }\in I$ and $\mathcal{M}(\overline{1})=1$;
(F2)	$\mathcal{M}(f\wedge g)=\mathcal{M}(f)\wedge \mathcal{M}(g)$ for all $f,g\in I^X$.

If $\mathcal{M}$ and $\mathcal{N}$ are fuzzy filters on X, then $\mathcal{M}$ is said to be finer than $\mathcal{N}$, denoted by $\mathcal{M}\le \mathcal{N}$, provided that $\mathcal{M}(f)\ge \mathcal{N}(f)$ for every $f\in I^X$. By $\mathcal{M}\nleqq \mathcal{N}$ we mean that $\mathcal{M}$ is not finer than $\mathcal{N}$. $\mathcal{M}\nleqq \mathcal{N}⟺\text{there is}f\in I^X\text{such that}\mathcal{M}(f)<\mathcal{N}(f)$.

A non-empty subset $\mathcal{F}$ of $I^X$ is called a prefilter on X (Lowen, Lowen), provided that the following conditions are fulfilled:

(1)	$\overline{0}\notin \mathcal{F}$;
(2)	$f,g\in \mathcal{F}$ implies $f\wedge g\in \mathcal{F}$;
(3)	$f\in \mathcal{F}$ and $f\le g$ imply $g\in \mathcal{F}$.

For each fuzzy filter $\mathcal{M}$ on X, the subset $\mathit{\alpha }\text{-}\mathrm{pr}\mathcal{M}$ of $I^X$ defined by:$$\mathit{\alpha }\text{-}\mathrm{pr}\mathcal{M}=\{f\in I^X\mid \mathcal{M}(f)\ge \mathit{\alpha }\}$$

is a prefilter on X.

Proposition 2.1

(Gähler, 1995a) There is a one-to-one correspondence between fuzzy filters $\mathcal{M}$ on X and the families ${({\mathcal{M}}_{\mathit{\alpha }})}_{\mathit{\alpha }\in I_0}$ of prefilters on X which fulfill the following conditions:

(1)	$f\in {\mathcal{M}}_{\mathit{\alpha }}$ implies $\mathit{\alpha }\le supf$.
(2)	$0<\mathit{\alpha }\le \mathit{\beta }$ implies ${\mathcal{M}}_{\mathit{\alpha }}\supseteq {\mathcal{M}}_{\mathit{\beta }}$.
(3)	For each $\mathit{\alpha }\in I_0$ with $\underset{0<\mathit{\beta }<\mathit{\alpha }}{\bigvee }\mathit{\beta }=\mathit{\alpha }$, we have $\bigcap _{0<\mathit{\beta }<\mathit{\alpha }}{\mathcal{M}}_{\mathit{\beta }}={\mathcal{M}}_{\mathit{\alpha }}$.

This correspondence is given by ${\mathcal{M}}_{\mathit{\alpha }}=$$\mathit{\alpha }$-pr $\mathcal{M}$ for all $\mathit{\alpha }\in I_0$ and $\mathcal{M}(f)=\underset{g\in {\mathcal{M}}_{\mathit{\alpha }},g\le f}{\bigvee }\mathit{\alpha }$ for all $f\in I^X$.

Proposition 2.2

(Eklund, 1992) Let A be a set of fuzzy filters on X. Then, the following are equivalent.

(1)	The infimum $\underset{\mathcal{M}\in A}{\bigwedge }\mathcal{M}$ of A with respect to the finer relation of fuzzy filters exists,
(2)	For each non-empty finite subset $\{{\mathcal{M}}_1,\dots ,{\mathcal{M}}_n\}$ of A, we have ${\mathcal{M}}_1(f_1)\wedge \cdots \wedge {\mathcal{M}}_n(f_n)\le sup(f_1\wedge \cdots \wedge f_n)$ for all $f_1,\dots ,f_n\in I^X$,
(3)	For each $\mathit{\alpha }\in I_0$ and each non-empty finite subset $f_1,\dots ,f_n$ of $\bigcup _{\mathcal{M}\in A}\mathit{\alpha }\text{-pr}\mathcal{M}$, we have $\mathit{\alpha }\le sup(f_1\wedge \cdots \wedge f_n)$.

Recall that $(\underset{\mathcal{M}\in A}{\bigwedge }\mathcal{M})(f)=\underset{\mathcal{M}\in A}{\bigvee }\mathcal{M}(f)$ and $(\underset{\mathcal{M}\in A}{\bigvee }\mathcal{M})(f)=\underset{\mathcal{M}\in A}{\bigwedge }\mathcal{M}(f)$ for all $f\in I^X$.

Fuzzy neighbourhood filters. For each fuzzy topological space $(X,\mathit{\tau })$ and each $x\in X$, the mapping ${\mathcal{N}}_x:I^X\to I$ defined by (Gähler, Gahler):$${\mathcal{N}}_x(\mathit{\lambda })={\mathrm{int}}_{\mathit{\tau }}\mathit{\lambda }(x)\text{for all}\mathit{\lambda }\in I^X$$

is a fuzzy filter on X, called the fuzzy neighbourhood filter of the space $(X,\mathit{\tau })$ at the point x, and for short is called a fuzzy neighbourhood filter at x. The mapping $\dot{x}:I^X\to I$ is defined by $\dot{x}(\mathit{\lambda })=\mathit{\lambda }(x)$ for all $\mathit{\lambda }\in I^X$. The fuzzy neighbourhood filters fulfil the following conditions:

(1)	$\dot{x}\le {\mathcal{N}}_x$ holds for all $x\in X$;
(2)	$({\mathcal{N}}_x)({\mathrm{int}}_{\mathit{\tau }}f)=({\mathcal{N}}_x)(f)$ for all $x\in X$ and $f\in I^X$.

A fuzzy filter $\mathcal{M}$ is said to converge to $x\in X$, denoted by $\mathcal{M}x$, if $\mathcal{M}\le {\mathcal{N}}_x$ (Gähler, 1995b).

The fuzzy neighbourhood filter ${\mathcal{N}}_F$ at an ordinary subset F of X is the fuzzy filter on X defined in Bayoumi and Ibedou (2002b), by means of ${\mathcal{N}}_x$, $x\in F$ as:$${\mathcal{N}}_F=\underset{x\in F}{\bigvee }{\mathcal{N}}_x.$$

The fuzzy filter $\dot{F}$ is defined by$$\dot{F}=\underset{x\in F}{\bigvee }\dot{x}.$$$\dot{F}\le {\mathcal{N}}_F$ holds for all $F\subseteq X$. Also, recall that the fuzzy filter $\dot{\mathit{\lambda }}$ and the fuzzy neighbourhood filter ${\mathcal{N}}_{\mathit{\lambda }}$ at a fuzzy subset $\mathit{\lambda }$ of X are defined by(1) $$\begin{array}{c}\hfill \dot{\mathit{\lambda }}=\underset{0<\mathit{\lambda }(x)}{\bigvee }\dot{x}\text{and}{\mathcal{N}}_{\mathit{\lambda }}=\underset{0<\mathit{\lambda }(x)}{\bigvee }{\mathcal{N}}_x,\end{array}$$(1)

respectively. $\dot{\mathit{\lambda }}\le {\mathcal{N}}_{\mathit{\lambda }}$ holds for all $\mathit{\lambda }\in I^X$ (Bayoumi & Ibedou, 2004).

For each fuzzy topological space $(X,\mathit{\tau })$ the closure operator $\mathrm{cl}$ which assigns to each fuzzy filter $\mathcal{M}$ on X, the fuzzy filter $\mathrm{cl}\mathcal{M}$ is defined by(2) $$\begin{array}{c}\hfill \mathrm{cl}\mathcal{M}(f)=\underset{{\mathrm{cl}}_{\mathit{\tau }}g\le f}{\bigvee }\mathcal{M}(g).\end{array}$$(2) $\mathrm{cl}\mathcal{M}$ is called the closure of $\mathcal{M}$. $\mathrm{cl}$ is isotone, hull and idempotent operator, that is for all fuzzy filters $\mathcal{M}$ and $\mathcal{N}$ on X, we have (Gähler, 1995b):(3) $$\begin{array}{c}\hfill \mathcal{M}\le \mathcal{N}\text{implies}\mathrm{cl}\mathcal{M}\le \mathrm{cl}\mathcal{N},\end{array}$$(3) (4) $$\begin{array}{c}\hfill \mathcal{M}\le \mathrm{cl}\mathcal{M},\end{array}$$(4)

3. Neighbourhood systems

Definition 3.1

A family ${({\mathcal{N}}_x^{\mathit{\alpha }})}_{x\in X}$ of fuzzy sets ${\mathcal{N}}_x^{\mathit{\alpha }}$ is said to be a neighbourhood system with grade $\mathit{\alpha }\in I_0$ on X if it satisfies the following conditions:

(Nb1)	For all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$, we have $\mathit{\alpha }\le f(x)$,
(Nb2)	$\overline{1}\in {\mathcal{N}}_x^{\mathit{\alpha }}$,
(Nb3)	$f,g\in {\mathcal{N}}_x^{\mathit{\alpha }}$ implies that $f\wedge g\in {\mathcal{N}}_x^{\mathit{\alpha }}$,
(Nb4)	$f\in {\mathcal{N}}_x^{\mathit{\alpha }}$, $f\le g$ imply that $g\in {\mathcal{N}}_x^{\mathit{\alpha }}$,
(Nb5)	If $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$, then there is $g\in {\mathcal{N}}_x^{\mathit{\alpha }}$, such that for all $y\in X$ with $0<g(y)$, we have $f\in {\mathcal{N}}_y^{\mathit{\alpha }}$.

Lemma 3.2

These families of prefilters ${({\mathcal{N}}_x^{\mathit{\alpha }})}_{\mathit{\alpha }\in I_0}$ at $x\in X$ satisfy the following conditions:

(Pr1)	$f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ implies that $\mathit{\alpha }\le \sup f$,
(Pr2)	$0<\mathit{\beta }\le \mathit{\alpha }$ implies that ${\mathcal{N}}_x^{\mathit{\alpha }}\subseteq {\mathcal{N}}_x^{\mathit{\beta }}$,
(Pr3)	For every $\mathit{\alpha }\in I_0$ with $\underset{0<\mathit{\beta }<\mathit{\alpha }}{\bigvee }\mathit{\beta }=\mathit{\alpha }$, we have $\bigcap _{0<\mathit{\beta }<\mathit{\alpha }}{\mathcal{N}}_x^{\mathit{\beta }}={\mathcal{N}}_x^{\mathit{\alpha }}$.

Proof

Clear. $\square$

Remark 3.3

For any subset A of X, let us define ${\mathcal{N}}_A^{\mathit{\alpha }}$ by ${\mathcal{N}}_A^{\mathit{\alpha }}=\bigcap _{x\in A}{\mathcal{N}}_x^{\mathit{\alpha }}$, that is $f\in {\mathcal{N}}_A^{\mathit{\alpha }}$ iff $\mathit{\alpha }\le \underset{x\in A}{\bigwedge }{\mathcal{N}}_x(f)$ iff $\mathit{\alpha }\le {\mathcal{N}}_A(f)$. ${\mathcal{N}}_x^{\mathit{\alpha }}={\mathcal{N}}_{\{x\}}^{\mathit{\alpha }}$, ${\mathcal{N}}_x^{\mathit{\alpha }}\cap {\mathcal{N}}_x^{\mathit{\beta }}={\mathcal{N}}_x^{\mathit{\alpha }\vee \mathit{\beta }}$, ${\mathcal{N}}_x^{\mathit{\alpha }}\cup {\mathcal{N}}_x^{\mathit{\beta }}={\mathcal{N}}_x^{\mathit{\alpha }\wedge \mathit{\beta }}$, $\bigcap _j{\mathcal{N}}_x^{{\mathit{\alpha }}_j}={\mathcal{N}}_x{}^{\underset{j}{\bigvee }{\mathit{\alpha }}_j}$, $\bigcup _j{\mathcal{N}}_x^{{\mathit{\alpha }}_j}={\mathcal{N}}_x{}^{\underset{j}{\bigwedge }{\mathit{\alpha }}_j}$. For all $\mathit{\alpha }\ne \mathit{\beta }$ in $I_0$, we have ${\mathcal{N}}_x^{\mathit{\alpha }}\ne {\mathcal{N}}_x^{\mathit{\beta }}$.

For any $\mathit{\alpha },\mathit{\beta },\mathit{\gamma }\in I_0$, we have ${\mathcal{N}}_x^{\mathit{\alpha }}\subseteq {\mathcal{N}}_x^{\mathit{\alpha }}$, ${\mathcal{N}}_x^{\mathit{\alpha }}\subseteq {\mathcal{N}}_x^{\mathit{\beta }}$ and ${\mathcal{N}}_x^{\mathit{\beta }}\subseteq {\mathcal{N}}_x^{\mathit{\alpha }}$ implies that $\mathit{\alpha }=\mathit{\beta }$, ${\mathcal{N}}_x^{\mathit{\alpha }}\subseteq {\mathcal{N}}_x^{\mathit{\beta }}$ and ${\mathcal{N}}_x^{\mathit{\beta }}\subseteq {\mathcal{N}}_x^{\mathit{\gamma }}$ implies that ${\mathcal{N}}_x^{\mathit{\alpha }}\subseteq {\mathcal{N}}_x^{\mathit{\gamma }}$. Also, for all $\mathit{\alpha }\ne \mathit{\beta }\in I_0$, we have either ${\mathcal{N}}_x^{\mathit{\alpha }}\subseteq {\mathcal{N}}_x^{\mathit{\beta }}$ or ${\mathcal{N}}_x^{\mathit{\beta }}\subseteq {\mathcal{N}}_x^{\mathit{\alpha }}$.

$(\mathit{\alpha })$ fuzzy open sets, fuzzy open sets.

Let us define an $(\mathit{\alpha })$ fuzzy open set as follows:(5) $$\begin{array}{c}\hfill \mathit{\alpha }\le \mathit{\tau }(f)\mathsf{iff}\mathrm{for}\mathrm{all}x\in X\mathrm{there}\mathrm{is}\mathit{\alpha }\in I_0\mathrm{such}\mathrm{that}f\in {\mathcal{N}}_x^{\mathit{\alpha }}\mathrm{and}f(x)\le \mathit{\alpha }.\end{array}$$(5)

An $(\mathit{\alpha })$ fuzzy closed set is the complement of an $(\mathit{\alpha })$ fuzzy open set.

A set $f\in I^X$ is said to be fuzzy open if it is $(\mathit{\alpha })$ fuzzy open for all $\mathit{\alpha }\in I_0$. In other words, if for all $x\in X$ and for all $\mathit{\alpha }\in I_0$, we have $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $f(x)\le \mathit{\alpha }.$

It is called a fuzzy closed if it is the complement of a fuzzy open set. These notations are restricted to the usual open and closed sets in fuzzy topology and usual topology.

Starting from a neighbourhood system ${({\mathcal{N}}_x^{\mathit{\alpha }})}_{x\in X}$ with grade $\mathit{\alpha }\in I_0$, we can define an interior operator and a closure operator as follows:(6) $$\begin{array}{c}\hfill \mathrm{int}f(x)=\underset{g\in {\mathcal{N}}_x^{\mathit{\alpha }}}{\bigvee }\underset{0<g(y)}{\bigwedge }f(y),\end{array}$$(6) (7) $$\begin{array}{c}\hfill \mathrm{cl}f(x)=\underset{g\in {\mathcal{N}}_x^{\mathit{\alpha }}}{\bigwedge }\underset{0<g(y)}{\bigvee }f(y).\end{array}$$(7) For every $x\in X$, ${\mathcal{N}}_x^{\mathit{\alpha }}$ satisfying (Nb1) to (Nb4) is exactly a prefilter on X of all neighbourhoods of $x\in X$ with grade $\mathit{\alpha }\in I_0$. That is, ${({\mathcal{N}}_x^{\mathit{\alpha }})}_{x\in X}$ is a family of prefilters with grade $\mathit{\alpha }\in I_0$ at every $x\in X$ constructing after adding condition (Nb5) a neighbourhood system on X with grade $\mathit{\alpha }\in I_0$. The pair $(X,{({\mathcal{N}}_x^{\mathit{\alpha }})}_{x\in X})$ is called a neighbourhood space with a grade $\mathit{\alpha }\in I_0$.

From Lemma 2.1 and from the correspondence given in Proposition 2.1 between the fuzzy filters and the families satisfying the conditions (Pr1) to (Pr3), we can say this family ${({\mathcal{N}}_x^{\mathit{\alpha }})}_{\mathit{\alpha }\in I_0}$ is a representation of the fuzzy neighbourhood filter ${\mathcal{N}}_x$ as a family of prefilters. This is given by the following two conditions (Nb) and (Pr):

(Pr)	${\mathcal{N}}_x(f)=\underset{g\in {\mathcal{N}}_x^{\mathit{\alpha }},g\le f}{\bigvee }\mathit{\alpha }$ for all $f\in I^X$.
(Nb)	${\mathcal{N}}_x^{\mathit{\alpha }}=\{f\in I^X\mid \mathit{\alpha }\le {\mathcal{N}}_x(f)\}$.

Denote the subset ${\mathcal{N}}_x^{\mathit{\alpha }}\subseteq I^X$ as the fuzzy neighbourhoods with grade $\mathit{\alpha }\in I_0$ of $x\in X$.

Clearly, both the interior operator and closure operator satisfy the common axioms of interior operator and closure operator, respectively. A fuzzy topology on X could be generated by this interior operator given by (2.2) or this closure operator given by (2.3), using the properties of ${\mathcal{N}}_x^{\mathit{\alpha }}$ stated in (Nb1)—(Nb5). That fuzzy topology is exactly the fuzzy topology $\mathit{\tau }$ associated with the fuzzy neighbourhood filters ${\mathcal{N}}_x$ given by an interior operation as in (2.2) so that$${\mathcal{N}}_x(f)={\mathrm{int}}_{\mathit{\tau }}f(x)\mathrm{for}\mathrm{all}f\in I^X.$$

Also, we can consider(8) $$\begin{array}{c}\hfill {\mathcal{N}}_x^{\mathit{\alpha }}=\{f\in I^X\mid \mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}f(x)\}\end{array}$$(8)

and then, (2.1) for an $(\mathit{\alpha })$ fuzzy open set could be rewritten as(9) $$\begin{array}{c}\hfill \mathit{\alpha }\le \mathit{\tau }(f)\mathsf{iff}\mathrm{for}\mathrm{all}x\in X,\mathrm{there}\mathrm{is}\mathit{\alpha }\in I_0\mathrm{so}\mathrm{that}\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}f(x),f(x)\le \mathit{\alpha }.\end{array}$$(9)

That is, from a neighbourhood system of graded neighbourhoods, we can deduce interior operation by which it is introduced a graded fuzzy topology and the converse is true.

From (1.2) and (1.4) for all $x\in X$ and all $\mathit{\alpha }\in I_0$, we can define $\mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}$ by(10) $$\begin{array}{c}\hfill \mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}=\{f\in I^X\mid \mathit{\alpha }\le \mathrm{cl}{\mathcal{N}}_x(f)\},\end{array}$$(10)

and equivalently,(11) $$\begin{array}{c}\hfill \mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}=\{f\in I^X\mid \mathrm{there}\mathrm{is}h\in {\mathcal{N}}_x^{\mathit{\alpha }},\mathrm{cl}h\le f\}.\end{array}$$(11)

For all $x\in X$ and all $\mathit{\alpha }\in I_0$, we have $\mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}\subseteq {\mathcal{N}}_x^{\mathit{\alpha }}$.

Definition 3.4

Let $(X,{\mathit{\tau }}_1)$ and $(Y,{\mathit{\tau }}_2)$ be fuzzy topological spaces, and $f:X\to Y$ a map. Then, for some $\mathit{\alpha }\in I_0$, f is called ($\mathit{\alpha }$) fuzzy continuous if for all ($\mathit{\alpha }$) fuzzy open set $\mathit{\mu }$ with respect to ${\mathit{\tau }}_2$, we have $f^{-1}(\mathit{\mu })$ is an ($\mathit{\alpha }$) fuzzy open set with respect to ${\mathit{\tau }}_1$ for all $\mathit{\mu }\in I^Y$.

f is called fuzzy continuous if for all fuzzy open set $\mathit{\mu }$ with respect to ${\mathit{\tau }}_2$, we have $f^{-1}(\mathit{\mu })$ is a fuzzy open set with respect to ${\mathit{\tau }}_1$ for all $\mathit{\mu }\in I^Y$.

Definition 3.5

Let $(X,{\mathit{\tau }}_1)$ and $(Y,{\mathit{\tau }}_2)$ be fuzzy topological spaces. Then, the mapping $f:(X,{\mathit{\tau }}_1)\to (Y,{\mathit{\tau }}_2)$ is called ($\mathit{\alpha }$) fuzzy open (($\mathit{\alpha }$) fuzzy closed) mapping if the image f(g) of the ($\mathit{\alpha }$) fuzzy open (($\mathit{\alpha }$) fuzzy closed) set g with respect to ${\mathit{\tau }}_1$ is ($\mathit{\alpha }$) fuzzy open (($\mathit{\alpha }$) fuzzy closed) set with respect to ${\mathit{\tau }}_2$.

The mapping $f:(X,{\mathit{\tau }}_1)\to (Y,{\mathit{\tau }}_2)$ is called fuzzy open (fuzzy closed) mapping if the image f(g) of the fuzzy open (fuzzy closed) set g with respect to ${\mathit{\tau }}_1$ is fuzzy open (fuzzy closed) set with respect to ${\mathit{\tau }}_2$.

Now, we define the continuity locally at a point $x_0\in X$ between two fuzzy topological spaces using these graded neighbourhoods.

Definition 3.6

Let $(X,\mathit{\tau })$ and $(Y,\mathit{\sigma })$ be two fuzzy topological spaces. Then, the mapping $f:(X,\mathit{\tau })\to (Y,\mathit{\sigma })$ is called $(\mathit{\alpha })$ fuzzy continuous at a point $x_0$   provided that for all $g\in {\mathcal{N}}_{f(x_0)}^{\mathit{\alpha }}$,

there exists $h\in {\mathcal{N}}_{x_0}^{\mathit{\alpha }}$ such that $h\le f^{-1}(g)$ for some $\mathit{\alpha }\in I_0$. f is $(\mathit{\alpha })$ fuzzy continuous if it is $(\mathit{\alpha })$ fuzzy continuous at every $x\in X$. f is an fuzzy continuous if it is $(\mathit{\alpha })$ fuzzy continuous for all $\mathit{\alpha }\in I_0$.

This is an equivalent definition with Definition 2.2 for the $(\mathit{\alpha })$ fuzzy continuous mapping and fuzzy continuous mapping.

4. $(\mathit{\alpha },\mathit{\beta })T_0$-spaces and $(\mathit{\alpha },\mathit{\beta })T_1$-spaces

This section is devoted to introduce the notions of $T_0$-spaces and $T_1$-spaces using the notion of $\mathit{\alpha }$-neighbourhoods at ordinary points. We will introduce different equivalent definitions,

and we show that these notions are good extensions in sense of Lowen (1978]).

Definition 4.1

A fuzzy topological space $(X,\mathit{\tau })$ is called an $(\mathit{\alpha },\mathit{\beta })T_0$-space if for all $x\ne y$ in X, there exists $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ such that $f(y)<\mathit{\alpha }$; $\mathit{\alpha }\in I_0$ or there exists $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $g(x)<\mathit{\beta }$; $\mathit{\beta }\in I_0$.

Definition 4.2

A fuzzy topological space $(X,\mathit{\tau })$ is called an $(\mathit{\alpha },\mathit{\beta })T_1$-space if for all $x\ne y$ in X there exist $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $f(y)<\mathit{\alpha }$ and $g(x)<\mathit{\beta }$; $\mathit{\alpha },\mathit{\beta }\in I_0$.

Example 4.3

Let $X=\{x,y\}$, and $$\mathit{\tau }(f)=\left\{\begin{array}{cc}1\hfill & \mathrm{at}\overline{0}\mathrm{or}\overline{1}\hfill \\ \frac{1}{3}\hfill & \mathrm{at}{x}_{\frac{1}{2}}\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$Taking $\mathit{\alpha }=\frac{1}{3}$, we get that there is $f=x_{\frac{1}{2}}$ in ${\mathcal{N}}_x^{\mathit{\alpha }}$ such that $f(y)<\mathit{\alpha }$. For all $\mathit{\alpha }\in I_0$, we can not find any f in ${\mathcal{N}}_y^{\mathit{\alpha }}$ such that $f(x)<\mathit{\alpha }$. That is, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })T_0$-space.

Example 4.4

Let $X=\{x,y\}$, and$$\mathit{\tau }(f)=\left\{\begin{array}{cc}1\hfill & \mathrm{at}\overline{0}\mathrm{or}\overline{1}\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Only there is $f=\overline{1}$ which is a graded neighbourhood but for both of x, y. Hence, for all $\mathit{\alpha }\in I_0$, ${\mathcal{N}}_x^{\mathit{\alpha }}={\mathcal{N}}_y^{\mathit{\alpha }}$ and therefore, $(X,\mathit{\tau })$ is not $(\mathit{\alpha },\mathit{\beta })T_0$-space.

Proposition 4.5

Every $(\mathit{\alpha },\mathit{\beta })$$T_1$-space is an $(\mathit{\alpha },\mathit{\beta })$$T_0$-space.

Proof

Clear. $\square$

Example 3.1 is an $(\mathit{\alpha },\mathit{\beta })$$T_0$-space but not $(\mathit{\alpha },\mathit{\beta })$$T_1$-space.

Example 4.6

Let $X=\{x,y\}$, and $$\mathit{\tau }(f)=\left\{\begin{array}{cc}1\hfill & \mathrm{at}\overline{0}\mathrm{or}\overline{1}\hfill \\ \frac{1}{3}\hfill & \mathrm{at}{x}_{\frac{1}{2}}\hfill \\ \frac{1}{3}\hfill & \mathrm{at}{y}_{\frac{1}{2}}\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$Taking $\mathit{\alpha }=\frac{1}{3}$ and $\mathit{\beta }=\frac{1}{3}$, we get that there is $f=x_{\frac{1}{2}}$ in ${\mathcal{N}}_x^{\mathit{\alpha }}$ and $g=y_{\frac{1}{2}}$ in ${\mathcal{N}}_y^{\mathit{\beta }}$ such that $f(y)<\mathit{\alpha }$ and $g(x)<\mathit{\beta }$, for some $\mathit{\alpha },\mathit{\beta }\in I_0$. Hence, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })T_1$-space.

In the following theorems, there will be introduced some equivalent definitions for the $(\mathit{\alpha },\mathit{\beta })T_0$-spaces and $(\mathit{\alpha },\mathit{\beta })T_1$-spaces.

Theorem 4.7

Let $(X,\mathit{\tau })$ be a fuzzy topological space. Then, the following statements are equivalent.

(1)	$(X,\mathit{\tau })$ is $(\mathit{\alpha },\mathit{\beta })T_0$.
(2)	For all $x\ne y$ in X and for all $\mathit{\alpha }\in I_0$, ${\mathcal{N}}_x^{\mathit{\alpha }}\ne {\mathcal{N}}_y^{\mathit{\alpha }}$.
(3)	For all $x\ne y$ in X, there exists $f\in I^X$ such that $f(y)<\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}f(x)$; $\mathit{\alpha }\in I_0$ or there exists $g\in I^X$ such that $g(x)<\mathit{\beta }\le {\mathrm{int}}_{\mathit{\tau }}g(y)$; $\mathit{\beta }\in I_0$.
(4)	For all $x\ne y$ in X, there exists $f\in I^X$ such that $f(y)<{\mathrm{cl}}_{\mathit{\tau }}f(x)$ or there exists $g\in I^X$ such that $g(x)<{\mathrm{cl}}_{\mathit{\tau }}g(y)$.

Proof

$(1)\Rightarrow (2)$: From (1), there is $f\in I^X$ such that ${\mathrm{int}}_{\mathit{\tau }}f(y)\le f(y)<\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}f(x)$; $\mathit{\alpha }\in I_0$ and then, $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $f\notin {\mathcal{N}}_y^{\mathit{\alpha }}$. Hence, ${\mathcal{N}}_x^{\mathit{\alpha }}\ne {\mathcal{N}}_y^{\mathit{\alpha }}$; $\mathit{\alpha }\in I_0$ and thus, (2) holds.

$(2)\Rightarrow (3)$: There exists $f\in I^X$ such that ${\mathrm{int}}_{\mathit{\tau }}f(y)<\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}f(x)$; $\mathit{\alpha }\in I_0$ and then, for $g={\mathrm{int}}_{\mathit{\tau }}f$, we can say $g(y)<\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}g(x)$; $\mathit{\alpha }\in I_0$. The other case is similar and thus, (3) is satisfied.

$(3)\Rightarrow (4)$: From Equation 2.3, we get that $c{l}_{\mathit{\tau }}f)(x)\ge f(y)$ whenever ${\mathrm{int}}_{\mathit{\tau }}f(y)\le \mathit{\alpha }\le f(x)$, then (4) holds.

$(4)\Rightarrow (1)$: Since $f(y)<\underset{h\in {\mathcal{N}}_x^{\mathit{\alpha }}}{\bigwedge }\underset{0<h(z)}{\bigvee }f(z)=c{l}_{\mathit{\tau }}f(x)$ implies that z could not be y with $0<h(y)$ for all $h\in {\mathcal{N}}_x^{\mathit{\alpha }}$; $\mathit{\alpha }\in I_0$, which means that there is $h\in {\mathcal{N}}_x^{\mathit{\alpha }}$ such that $h(y)=0<\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}h(x)$; $\mathit{\alpha }\in I_0$. The other case is similar and thus, (1) holds. $\square$

Theorem 4.8

Let $(X,\mathit{\tau })$ be a fuzzy topological space. Then, the following statements are equivalent.

(1)	$(X,\mathit{\tau })$ is $(\mathit{\alpha },\mathit{\beta })T_1$.
(2)	For all $x\in X$, we have ${\mathrm{cl}}_{\mathit{\tau }}{x}_1=x_1$.
(3)	For all $x\ne y$ in X, there exist $f,g\in I^X$ such that $f(y)<\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}f(x)$ and $g(x)<\mathit{\beta }\le {\mathrm{int}}_{\mathit{\tau }}g(y)$; $\mathit{\alpha },\mathit{\beta }\in I_0$.
(4)	For all $x\ne y$ in X, there exist $f,g\in I^X$ such that $f(y)<{\mathrm{cl}}_{\mathit{\tau }}f(x)$ and $g(x)<{\mathrm{cl}}_{\mathit{\tau }}g(y)$.

Proof

$(1)\Rightarrow (2)$: Let $y\ne x$ in X. Then, ${\mathrm{cl}}_{\mathit{\tau }}{x}_1(y)=\underset{h\in {\mathcal{N}}_y^{\mathit{\alpha }}}{\bigwedge }\underset{0<h(z)}{\bigvee }{x}_1(z)$, which means for all $h\in {\mathcal{N}}_y^{\mathit{\alpha }}$, if $x_1(z)>0$ whenever $h(z)>0$, then ${\mathrm{cl}}_{\mathit{\tau }}{x}_1(y)>0$. From (1), we get that z could not be x with $0<h(x)$, that is, ${\mathrm{cl}}_{\mathit{\tau }}{x}_1(y)=0$ for all $y\ne x$. At x, it is clear that ${\mathrm{cl}}_{\mathit{\tau }}{x}_1(x)=1$. Hence, ${\mathrm{cl}}_{\mathit{\tau }}{x}_1=x_1$ for all $x\in X$, and (2) is fulfilled.

$(2)\Rightarrow (3)$: For all $x\ne y$ in X, we have ${\mathrm{cl}}_{\mathit{\tau }}{x}_1=x_1$ and ${\mathrm{cl}}_{\mathit{\tau }}{y}_1=y_1$. (2) means that ${\mathrm{cl}}_{\mathit{\tau }}{x}_1(y)=x_1(y)=0=\underset{h\in {\mathcal{N}}_y^{\mathit{\alpha }}}{\bigwedge }\underset{0<h(z)}{\bigvee }{x}_1(z)$, which means for all $h\in {\mathcal{N}}_y^{\mathit{\alpha }}$, z could not be x with $0<h(x)$, that is there is $\mathit{\alpha }\in I_0$ and there is $h\in {\mathcal{N}}_y^{\mathit{\alpha }}$ such that $h(x)=0<\mathit{\alpha }$ and then, $h(x)<\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}h(y)$. The other case is similar and therefore, (3) is fulfilled.

$(3)\Rightarrow (4)$: As in Theorem 4.1.

$(4)\Rightarrow (1)$: As in Theorem 4.1. $\square$

The next proposition shows that the separation axioms $(\mathit{\alpha },\mathit{\beta })T_0$ and $(\mathit{\alpha },\mathit{\beta })T_1$ are good extensions in sense of Lowen (1978).

Proposition 4.9

A topological space (X, T) is a $T_0$-space ($T_1$-space) if and only if the induced fuzzy topological space $(X,\mathit{\omega }(T))$ is an $(\mathit{\alpha },\mathit{\beta })T_0$-space ($(\mathit{\alpha },\mathit{\beta })T_1$-space).

Proof

Let (X, T) be $T_0$ ($T_1$) and let $x\ne y$. Then, there is a neighbourhood ${\mathcal{O}}_y\in T$ such that $x\notin {\mathcal{O}}_y$. Taking $f\in I^X$ such that ${\mathcal{O}}_y=s_{\mathit{\alpha }}f\in T$ for some $\mathit{\alpha }\in I_1$, we get $f(x)\le \mathit{\alpha }<{\mathrm{int}}_{\mathit{\omega }(T)}f(y)$, That is, $f(x)<\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\omega }(T)}f(y)$ for some $\mathit{\alpha }\in I_0$. Similarly, if there is a neighbourhood ${\mathcal{O}}_x\in T$ such that $y\notin {\mathcal{O}}_x$, we can find $g\in I^X$ such that ${\mathcal{O}}_x=s_{\mathit{\beta }}g\in T$ and $g(y)\le \mathit{\beta }<{\mathrm{int}}_{\mathit{\omega }(T)}g(x)$ for some $\mathit{\beta }\in I_1$, That is, $g(y)<\mathit{\beta }\le {\mathrm{int}}_{\mathit{\omega }(T)}g(x)$ for some $\mathit{\beta }\in I_0$. Hence, $(X,\mathit{\omega }(T))$ is an $(\mathit{\alpha },\mathit{\beta })T_0$-space ($(\mathit{\alpha },\mathit{\beta })T_1$).

Conversely, let $(X,\mathit{\omega }(T))$ be an $(\mathit{\alpha },\mathit{\beta })T_0$-space ($(\mathit{\alpha },\mathit{\beta })T_1$) and $x\ne y$. Then, there exists $f\in I^X$ such that $f(y)<\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\omega }(T)}f(x)$ for some $\mathit{\alpha }\in I_0$, which means $f(y)\le \mathit{\alpha }<{\mathrm{int}}_{\mathit{\omega }(T)}f(x)$ for some $\mathit{\alpha }\in I_1$, that is there is ${\mathrm{int}}_{\mathit{\omega }(T)}f\in I^X$ such that $s_{\mathit{\alpha }}{\mathrm{int}}_{\mathit{\omega }(T)}f={\mathcal{O}}_x\in T$ and $y\notin {\mathcal{O}}_x$. Similarly, the other case is proved. Hence, (X, T) is a $T_0$-space ($T_1$). $\square$

Proposition 4.10

Let $(X,\mathit{\tau })$ be an $(\mathit{\alpha },\mathit{\beta })$$T_0$-space ($(\mathit{\alpha },\mathit{\beta })T_1$) and let $\mathit{\sigma }$ be a fuzzy topology on X finer than $\mathit{\tau }$. Then, $(X,\mathit{\sigma })$ is also $(\mathit{\alpha },\mathit{\beta })$$T_0$-space ($(\mathit{\alpha },\mathit{\beta })T_1$-space).

Proof

$(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })$$T_0$-space ($(\mathit{\alpha },\mathit{\beta })T_1$) implying that there is $f\in I^X$ such that $\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}f(x)$ and $f(x)<\mathit{\alpha }$ or (and) there is $g\in I^X$ such that $\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}g(x)$ and $g(x)<\mathit{\alpha }$, which implies that $\mathit{\alpha }\le \mathit{\tau }(f)$ or (and)$\mathit{\alpha }\le \mathit{\tau }(g)$. Since $\mathit{\sigma }$ is finer than $\mathit{\tau }$, then $\mathit{\alpha }\le \mathit{\sigma }(f)$ or (and)$\mathit{\alpha }\le \mathit{\sigma }(g)$, and thus, $\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\sigma }}f(x)$ and $f(x)<\mathit{\alpha }$ or (and) $\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\sigma }}g(x)$ and $g(x)<\mathit{\alpha }$. Hence $(X,\mathit{\sigma })$ is an $(\mathit{\alpha },\mathit{\beta })$$T_0$-space ($(\mathit{\alpha },\mathit{\beta })T_1$). $\square$

5. $(\mathit{\alpha },\mathit{\beta })T_2$-spaces

Here, we introduce and study the Hausdorff separation axiom in fuzzy topological spaces.

Definition 4.11

An fuzzy topological space $(X,\mathit{\tau })$ is called an $(\mathit{\alpha },\mathit{\beta })T_2$-space if for all $x\ne y$ in X there exist $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$.

Proposition 4.12

Every $(\mathit{\alpha },\mathit{\beta })T_2$-space is an $(\mathit{\alpha },\mathit{\beta })T_1$-space.

Proof

Let $(X,\mathit{\tau })$ be an $(\mathit{\alpha },\mathit{\beta })T_2$-space but not $(\mathit{\alpha },\mathit{\beta })T_1$-space. That is, for $x\ne y$, we get for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ that $f(y)\ge \mathit{\alpha }$ for all $\mathit{\alpha }\in I_0$. Since for any $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ we have $g(y)\ge \mathit{\beta }$, then $(f\wedge g)(y)=f(y)\wedge g(y)\ge (\mathit{\alpha }\wedge \mathit{\beta })$ and thus, $\sup (f\wedge g)\ge (\mathit{\alpha }\wedge \mathit{\beta })$ which contradicts the axiom $(\mathit{\alpha },\mathit{\beta })T_2$. Hence, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })T_1$-space. $\square$

Example 4.13

Let $X=\{x,y\}$, and $$\mathit{\tau }(f)=\left\{\begin{array}{cc}1\hfill & \mathrm{at}\overline{0}\mathrm{or}\overline{1}\hfill \\ \frac{1}{5}\hfill & \mathrm{at}{x}_1\hfill \\ \frac{4}{5}\hfill & \mathrm{at}{x}_{\frac{1}{3}}\vee y_1\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$There are $f=x_1\in I^X$ and $g=x_{\frac{1}{3}}\vee y_1\in I^X$ such that, for $\mathit{\alpha }=\frac{1}{5}$ and $\mathit{\beta }=\frac{4}{5}$ in $I_0$, we get that $f=x_1\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g=x_{\frac{1}{3}}\vee y_1\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $f(y)=x_1(y)=0<\frac{1}{5}=\mathit{\alpha }$ and $g(x)=(x_{\frac{1}{3}}\vee y_1)(x)=\frac{1}{3}<\frac{4}{5}=\mathit{\beta }$. That is, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })$$T_1$-space. But for all fuzzy sets $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$, we get that $(\mathit{\alpha }\wedge \mathit{\beta })\le \sup (f\wedge g)$ and thus, $(X,\mathit{\tau })$ is not $(\mathit{\alpha },\mathit{\beta })$$T_2$-space.

Theorem 4.14

Let $(X,\mathit{\tau })$ be an fuzzy topological space. Then, the following statements are equivalent.

(1)	$(X,\mathit{\tau })$ is $(\mathit{\alpha },\mathit{\beta })T_2$.
(2)	For all fuzzy ultrafilter $\mathcal{M}$ on X and for all $x\ne y$, there is $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ such that $\mathcal{M}(f)<\mathit{\alpha }$; $\mathit{\alpha }\in I_0$ or there is $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $\mathcal{M}(g)<\mathit{\beta }$; $\mathit{\beta }\in I_0$.
(3)	For all fuzzy filter $\mathcal{M}$ on X and for all $x\ne y$, there is $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ such that $\mathcal{M}(f)<\mathit{\alpha }$; $\mathit{\alpha }\in I_0$ or there is $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $\mathcal{M}(g)<\mathit{\beta }$; $\mathit{\beta }\in I_0$.

Proof

$(1)\Rightarrow (2)$: Suppose that there is an fuzzy ultrafilter $\mathcal{M}$ on X such that $\mathcal{M}(f)\ge \mathit{\alpha }$ and $\mathcal{M}(g)\ge \mathit{\beta }$ for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$. That is, $\mathcal{M}(f\wedge g)=\mathcal{M}(f)\wedge \mathcal{M}(g)\ge \mathit{\alpha }\wedge \mathit{\beta }$, but in common we know that $\mathcal{M}(h)\le \sup h$ for all $h\in I^X$, which means that for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\alpha }}$, we have $\sup (f\wedge g)\ge (\mathit{\alpha }\wedge \mathit{\beta })$ and therefore, (1) implies (2) is satisfied.

$(2)\Rightarrow (3)$: Since for any fuzzy filter $\mathcal{M}$ on X we find a finer fuzzy ultrafilter $\mathcal{I}$ on X, that is $\mathcal{M}(f)\le \mathcal{I}(f)$ for all $f\in I^X$, then (2) implies that there is $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ such that $\mathcal{M}(f)\le \mathcal{I}(f)<\mathit{\alpha }$; $\mathit{\alpha }\in I_0$ or there is $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $\mathcal{M}(g)\le \mathcal{I}(g)<\mathit{\beta }$; $\mathit{\beta }\in I_0$. Thus, (3) holds.

$(3)\Rightarrow (1)$: Suppose for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$; $\mathit{\alpha },\mathit{\beta }\in I_0$ that $(\mathit{\alpha }\wedge \mathit{\beta })\le \sup (f\wedge g)$ and (3) is fulfilled. Then, for all fuzzy filter $\mathcal{M}$ on X, we have $\mathcal{M}(f)<\mathit{\alpha }$ or $\mathcal{M}(g)<\mathit{\beta }$; $\mathit{\alpha },\mathit{\beta }\in I_0$. Hence, $\mathcal{M}(f\wedge g)<(\mathit{\alpha }\wedge \mathit{\beta })\le \sup (f\wedge g)$, which means a contradiction to the common result that $\mathcal{M}(f\wedge g)\le \sup (f\wedge g)$ and therefore, $\mathcal{M}(f\wedge g)\le \sup (f\wedge g)<(\mathit{\alpha }\wedge \mathit{\beta })$. Thus, (1) is satisfied. $\square$

Example 4.15

Let $X=\{x,y\}$, and $$\mathit{\tau }(f)=\left\{\begin{array}{cc}1\hfill & \mathrm{at}\overline{0}\mathrm{or}\overline{1}\hfill \\ \frac{1}{4}\hfill & \mathrm{at}{x}_{\frac{1}{3}}\hfill \\ \frac{1}{4}\hfill & \mathrm{at}{y}_{\frac{1}{3}}\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$There are $f=x_{\frac{1}{3}}\in I^X$ and $g=y_{\frac{1}{3}}\in I^X$ such that for $\mathit{\alpha }=\frac{1}{4}$ and $\mathit{\beta }=\frac{1}{4}$ in $I_0$, we get that $f=x_{\frac{1}{3}}$ in ${\mathcal{N}}_x^{\mathit{\alpha }}$ and $g=y_{\frac{1}{3}}$ in ${\mathcal{N}}_y^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })=\frac{1}{4}>\sup (x_{\frac{1}{3}}\wedge y_{\frac{1}{3}})=0$ and thus, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })$$T_2$-space.

Proposition 4.16

A topological space (X, T) is a $T_2$-space if and only if the induced fuzzy topological space $(X,\mathit{\omega }(T))$ is an $(\mathit{\alpha },\mathit{\beta })T_2$-space.

Proof

Let $x\ne y$ in X. Then, there are ${\mathcal{O}}_x,{\mathcal{O}}_y\in T$ such that ${\mathcal{O}}_x\cap {\mathcal{O}}_y=\mathrm{\varnothing }$. Taking $f,g\in I^X$ such that $s_{\mathit{\alpha }}f={\mathcal{O}}_x,s_{\mathit{\beta }}g={\mathcal{O}}_y$ for some $\mathit{\alpha },\mathit{\beta }\in I_1$, then ${\mathrm{int}}_{\mathit{\omega }(T)}f(x)>\mathit{\alpha }$ and ${\mathrm{int}}_{\mathit{\omega }(T)}g(y)>\mathit{\beta }$; $\mathit{\alpha },\mathit{\beta }\in I_1$, that is ${\mathrm{int}}_{\mathit{\omega }(T)}f(x)\ge \mathit{\alpha }$ and ${\mathrm{int}}_{\mathit{\omega }(T)}g(y)\ge \mathit{\beta }$; $\mathit{\alpha },\mathit{\beta }\in I_0$ and then, $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $s_{\mathit{\alpha }}f\cap s_{\mathit{\beta }}g={\mathcal{O}}_x\cap {\mathcal{O}}_y=\mathrm{\varnothing }$, which means that there is no element $z\in X$ such that $(f\wedge g)(z)=f(z)\wedge g(z)\ge {\mathrm{int}}_{\mathit{\omega }(T)}f(z)\wedge {\mathrm{int}}_{\mathit{\omega }(T)}g(z)>(\mathit{\alpha }\wedge \mathit{\beta })$; $\mathit{\alpha },\mathit{\beta }\in I_1$, which means for all $z\in X$, we have $(f\wedge g)(z)\le (\mathit{\alpha }\wedge \mathit{\beta })$; $\mathit{\alpha },\mathit{\beta }\in I_1$. Hence, $\sup (f\wedge g)\le (\mathit{\alpha }\wedge \mathit{\beta })$; $\mathit{\alpha },\mathit{\beta }\in I_1$ and then, $\sup (f\wedge g)<(\mathit{\alpha }\wedge \mathit{\beta })$; $\mathit{\alpha },\mathit{\beta }\in I_0$ and thus, $(X,\mathit{\omega }(T))$ is an $(\mathit{\alpha },\mathit{\beta })T_2$-space.

Conversely, $x\ne y$ implies that there are $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that ${\mathrm{int}}_{\mathit{\omega }(T)}f(x)\wedge {\mathrm{int}}_{\mathit{\omega }(T)}g(y)\ge (\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$. That is, for $\mathit{\gamma }=\sup (f\wedge g)\in I_1$, we can say ${\mathrm{int}}_{\mathit{\omega }(T)}f\in \mathit{\omega }(T),x\in s_{\mathit{\gamma }}{\mathrm{int}}_{\mathit{\omega }(T)}f$ and ${\mathrm{int}}_{\mathit{\omega }(T)}g\in \mathit{\omega }(T),y\in s_{\mathit{\gamma }}{\mathrm{int}}_{\mathit{\omega }(T)}g$, which means that $s_{\mathit{\gamma }}{\mathrm{int}}_{\mathit{\omega }(T)}f={\mathcal{O}}_x\in T$, $s_{\mathit{\gamma }}{\mathrm{int}}_{\mathit{\omega }(T)}g={\mathcal{O}}_y\in T$ and moreover, ${\mathcal{O}}_x\cap {\mathcal{O}}_y=\mathrm{\varnothing }$ and thus, (X, T) is a $T_2$-space. (because if there is $z\in ({\mathcal{O}}_x\cap {\mathcal{O}}_y)$, then $(f\wedge g)(z)\ge {\mathrm{int}}_{\mathit{\omega }(T)}f(z)\wedge {\mathrm{int}}_{\mathit{\omega }(T)}g(z)>\mathit{\gamma }=\sup (f\wedge g)$ which is a contradiction). $\square$

Proposition 4.17

Let $(X,\mathit{\tau })$ be an $(\mathit{\alpha },\mathit{\beta })$$T_2$-space, and let $\mathit{\sigma }$ be an fuzzy topology on X finer than $\mathit{\tau }$. Then, $(X,\mathit{\sigma })$ is also an $(\mathit{\alpha },\mathit{\beta })$$T_2$-space.

Proof

Let $x\ne y\in X$. Then, there are $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$, that is $\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}f(x)$, $\mathit{\beta }\le {\mathrm{int}}_{\mathit{\tau }}g(y)$ and $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$, which means that $\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\sigma }}f(x)$, $\mathit{\beta }\le {\mathrm{int}}_{\mathit{\sigma }}g(y)$ and $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$ and thus, $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ in $(X,\mathit{\sigma })$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$. Hence, $(X,\mathit{\sigma })$ is an $(\mathit{\alpha },\mathit{\beta })$$T_2$-space. $\square$

6. $(\mathit{\alpha },\mathit{\beta })T_3$-spaces and $(\mathit{\alpha },\mathit{\beta })T_4$-spaces

In this section, we use fuzzy neighbourhood filters at ordinary sets to define the notions of $(\mathit{\alpha },\mathit{\beta })T_3$-spaces and $(\mathit{\alpha },\mathit{\beta })$$T_4$-spaces.

Definition 5.1

A fuzzy topological space $(X,\mathit{\tau })$ is called $(\mathit{\alpha },\mathit{\beta })$ regular if for all $F={\mathrm{cl}}_{\mathit{\tau }}F$ in P(X) and $x\notin F$, there exist $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_F^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$.

Definition 5.2

A fuzzy topological space $(X,\mathit{\tau })$ is called $(\mathit{\alpha },\mathit{\beta })T_3$-space if it is regular and $(\mathit{\alpha },\mathit{\beta })T_1$.

Definition 5.3

A fuzzy topological space $(X,\mathit{\tau })$ is called normal if for all $F_1={\mathrm{cl}}_{\mathit{\tau }}{F}_1,F_2={\mathrm{cl}}_{\mathit{\tau }}{F}_2\in P(X)$ with $F_1\cap F_2=\mathrm{\varnothing }$, there exist $f\in {\mathcal{N}}_{F_1}^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_{F_2}^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha }\wedge \mathit{\beta }\in I_0$.

Definition 5.4

A fuzzy topological space $(X,\mathit{\tau })$ is called $(\mathit{\alpha },\mathit{\beta })T_4$ if it is normal and $(\mathit{\alpha },\mathit{\beta })T_1$.

Proposition 5.5

Every $(\mathit{\alpha },\mathit{\beta })T_3$-space is an $(\mathit{\alpha },\mathit{\beta })T_2$-space.

Proof

Let $x\ne y$ in X. $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })T_1$-space meaning that ${\mathrm{cl}}_{\mathit{\tau }}\{x\}=\{x\}$ for each $x\in X$. Now, ${\mathrm{cl}}_{\mathit{\tau }}\{y\}=\{y\}$, $x\notin \{y\}$, and $(X,\mathit{\tau })$ is regular implying that there are $f\in {\mathcal{N}}_x^{\mathit{\alpha }},g\in {\mathcal{N}}_y^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$. Hence, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })T_2$-space. $\square$

Theorem 5.6

For each fuzzy topological space $(X,\mathit{\tau })$, the following are equivalent.

(1)	$(X,\mathit{\tau })$ is regular.
(2)	For all $y\in F={\mathrm{cl}}_{\mathit{\tau }}F$ and $x\notin F$, we have ${\mathcal{N}}_x^{\mathit{\alpha }}⊈\mathrm{cl}{\mathcal{N}}_y^{\mathit{\alpha }}$ and ${\mathcal{N}}_y^{\mathit{\beta }}⊈\mathrm{cl}{\mathcal{N}}_x^{\mathit{\beta }}$ for all $y\in F$; $\mathit{\alpha },\mathit{\beta }\in I_0$.
(3)	For all $x\in X$ and all $\mathit{\alpha }\in I_0$, we have $\mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}={\mathcal{N}}_x^{\mathit{\alpha }}$.
(4)	For all $x\in X$, for all fuzzy filter $\mathcal{M}$ on X, for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$, and for all $\mathit{\alpha }\in I_0$, we have $\mathcal{M}(f)\ge \mathit{\alpha }$ implies $\mathrm{cl}\mathcal{M}(f)\ge \mathit{\alpha }$.

Proof

$(1)\Rightarrow (2)$: Let $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$; $\mathit{\alpha }\in I_0$. Suppose that $f\in \mathrm{cl}{\mathcal{N}}_y^{\mathit{\alpha }}$ for some $y\in F$, that is, there is $h\in {\mathcal{N}}_y^{\mathit{\alpha }}$ with ${\mathrm{cl}}_{\mathit{\tau }}h\le f$, which means that $f(y)\ge \mathit{\alpha }$. Since for all $g\in {\mathcal{N}}_y^{\mathit{\alpha }}$, we have $g(y)\ge \mathit{\alpha }$ for all $y\in F$; $\mathit{\alpha }\in I_0$, then $\sup (f\wedge g)\ge (f\wedge g)(y)\ge \mathit{\alpha }=(\mathit{\alpha }\wedge \mathit{\alpha })$ for some $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ for all $x\notin F$, and for all $g\in {\mathcal{N}}_y^{\mathit{\alpha }}$ for some $y\in F$; $\mathit{\alpha }\in I_0$, which contradicts (1) and therefore, $f\notin \mathrm{cl}{\mathcal{N}}_y^{\mathit{\alpha }}$ for all $y\in F$. Thus, ${\mathcal{N}}_x^{\mathit{\alpha }}⊈\mathrm{cl}{\mathcal{N}}_y^{\mathit{\alpha }}$ for all $y\in F$. The other case is similar and hence, (2) is satisfied.

$(2)\Rightarrow (3)$: From (2) we deduce that for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}\text{and}g\in {\mathcal{N}}_y^{\mathit{\beta }}$, we have $f\in \mathrm{cl}{\mathcal{N}}_y^{\mathit{\alpha }}\text{or}g\in \mathrm{cl}{\mathcal{N}}_x^{\mathit{\beta }}$ for all $\mathit{\alpha },\mathit{\beta }\in I_0$ implies $x=y$. Hence, for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$, $x\in X$, and all $\mathit{\alpha }\in I_0$, we get that $f\in \mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}$, which means that ${\mathcal{N}}_x^{\mathit{\alpha }}\subseteq \mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}$, but from that $\mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}\subseteq {\mathcal{N}}_x^{\mathit{\alpha }}$ for all $\mathit{\alpha }\in I_0$ and for all $x\in X$, we get that $\mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}={\mathcal{N}}_x^{\mathit{\alpha }}$ for all $\mathit{\alpha }\in I_0$ and for all $x\in X$ and thus, (3) holds.

$(3)\Rightarrow (4)$: Let $\mathcal{M}$ be a fuzzy filter on X with $\mathcal{M}(f)\ge \mathit{\alpha }$ for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $\mathit{\alpha }\in I_0$. From (3), $\mathcal{M}(f)\ge \mathit{\alpha }$ for all $f\in \mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}$ and $\mathit{\alpha }\in I_0$ and then, $\mathrm{cl}\mathcal{M}(f)\ge \mathit{\alpha }$ for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $\mathit{\alpha }\in I_0$ and thus, (4) is fulfilled.

$(4)\Rightarrow (1)$: Consider $\mathcal{M}={\mathcal{N}}_x$ in (4), we get that $\mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}={\mathcal{N}}_x^{\mathit{\alpha }}$ for all $x\in X$ and all $\mathit{\alpha }\in I_0$. Now, for $y\in F={\mathrm{cl}}_{\mathit{\tau }}F$ and $x\ne y$, we get for all $f\in {\mathcal{N}}_x^{\mathit{\alpha }}\text{and}g\in {\mathcal{N}}_y^{\mathit{\beta }}$ that $f\in \mathrm{cl}{\mathcal{N}}_x^{\mathit{\alpha }}\text{and}g\in \mathrm{cl}{\mathcal{N}}_y^{\mathit{\beta }}$, which means there are $h\in {\mathcal{N}}_x^{\mathit{\alpha }}\text{with}{\mathrm{cl}}_{\mathit{\tau }}h\le f$ and $k\in {\mathcal{N}}_y^{\mathit{\beta }}\text{with}{\mathrm{cl}}_{\mathit{\tau }}k\le g$. Choose $f={\mathrm{cl}}_{\mathit{\tau }}{\mathit{\chi }}_{F^c}\in {\mathcal{N}}_x^1$ and $g={\mathrm{cl}}_{\mathit{\tau }}({\mathrm{int}}_{\mathit{\tau }}{\mathit{\chi }}_F)\in {\mathcal{N}}_F^1$, then we can find $h={\mathit{\chi }}_{F^c}\in {\mathcal{N}}_x^1$ and $k={\mathrm{int}}_{\mathit{\tau }}{\mathit{\chi }}_F\in {\mathcal{N}}_F^1$ such that $(\mathit{\alpha }\wedge \mathit{\beta })=1>0=\mathrm{s}up({\mathit{\chi }}_{F^c}\wedge {\mathrm{int}}_{\mathit{\tau }}{\mathit{\chi }}_F)=\sup (h\wedge k)$, and thus, for all $F={\mathrm{cl}}_{\mathit{\tau }}F\in X$ a nd $x\notin F$, there exist $h\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $k\in {\mathcal{N}}_F^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (h\wedge k)$; $\mathit{\alpha },\mathit{\beta }\in I_0$, and therefore, (1) is satisfied. $\square$

Theorem 5.7

Let $(X,\mathit{\tau })$ be a fuzzy topological space. Then, the following are equivalent.

(1)	$(X,\mathit{\tau })$ is normal.
(2)	For all $F_1={\mathrm{cl}}_{\mathit{\tau }}{F}_1,F_2={\mathrm{cl}}_{\mathit{\tau }}{F}_2\in P(X)$ with $F_1\cap F_2=\mathrm{\varnothing }$, we have ${\mathcal{N}}_x^{\mathit{\alpha }}⊈\mathrm{cl}{\mathcal{N}}_y^{\mathit{\alpha }}$ and ${\mathcal{N}}_y^{\mathit{\beta }}⊈\mathrm{cl}{\mathcal{N}}_x^{\mathit{\beta }}$ for all $x\in F_1\text{and}y\in F_2$; $\mathit{\alpha },\mathit{\beta }\in I_0$.
(3)	For all $F={\mathrm{cl}}_{\mathit{\tau }}F\in P(X)$, and all $\mathit{\alpha }\in I_0$, we have $\mathrm{cl}{\mathcal{N}}_F^{\mathit{\alpha }}={\mathcal{N}}_F^{\mathit{\alpha }}$.
(4)	For all $F={\mathrm{cl}}_{\mathit{\tau }}F\in P(X)$, for all fuzzy filters $\mathcal{M}$ on X, for all $f\in {\mathcal{N}}_F^{\mathit{\alpha }}$, and for all $\mathit{\alpha }\in I_0$, we have $\mathcal{M}(f)\ge \mathit{\alpha }$ implies $\mathrm{cl}\mathcal{M}(f)\ge \mathit{\alpha }$.

Proof

Similar to the Theorem 6.1. $\square$

Proposition 5.8

Every $(\mathit{\alpha },\mathit{\beta })T_4$-space is an $(\mathit{\alpha },\mathit{\beta })T_3$-space.

Proof

Let $x\notin F={\mathrm{cl}}_{\mathit{\tau }}F$ in X. Since $(X,\mathit{\tau })$ is $(\mathit{\alpha },\mathit{\beta })T_4$, then it is $(\mathit{\alpha },\mathit{\beta })T_1$, which means that ${\mathrm{cl}}_{\mathit{\tau }}\{x\}=\{x\}$ for all $x\in X$, which implies that we have $F_1=\{x\}={\mathrm{cl}}_{\mathit{\tau }}\{x\}$ and $F_2=F$ with $F_1\cap F_2=\mathrm{\varnothing }$. Hence, there are $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_F^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$ and thus, $(X,\mathit{\tau })$ is regular and it is $(\mathit{\alpha },\mathit{\beta })T_1$. Therefore, $(X,\mathit{\tau })$ is $(\mathit{\alpha },\mathit{\beta })T_3$. $\square$

Example 5.9

Let $X=\{x,y\}$, and $$\mathit{\tau }(f)=\left\{\begin{array}{cc}1\hfill & \mathrm{at}\overline{0}\mathrm{or}\overline{1}\hfill \\ \frac{1}{2}\hfill & \mathrm{at}{x}_1\hfill \\ \frac{1}{3}\hfill & \mathrm{at}{y}_{\frac{1}{2}}\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$We notice that $\{y\}$ is a closed set and $x\notin \{y\}$. Then, there are $f=x_1\in I^X$ and $g=y_{\frac{1}{2}}\in I^X$ such that for $\mathit{\alpha }=\frac{1}{2}$ and $\mathit{\beta }=\frac{1}{3}$ in $I_0$, we get that $f=x_1$ in ${\mathcal{N}}_x^{\mathit{\alpha }}$ and $g=y_{\frac{1}{2}}$ in ${\mathcal{N}}_{\{y\}}^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })=\frac{1}{3}>\sup (x_1\wedge y_{\frac{1}{2}})=0$ and thus, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })$ regular space. Also, it is an $(\mathit{\alpha },\mathit{\beta })$$T_1$-space. Hence, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })$$T_3$-space

Example 6.2

Let $X=\{x,y\}$, and $$\mathit{\tau }(f)=\left\{\begin{array}{cc}1\hfill & \mathrm{at}\overline{0}\mathrm{or}\overline{1}\hfill \\ \frac{1}{2}\hfill & \mathrm{at}{x}_1\hfill \\ \frac{1}{2}\hfill & \mathrm{at}{y}_1\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$We see that $\{x\}$ and $\{y\}$ are disjoint closed subsets of X. Then, there are $f=x_1\in I^X$ and $g=y_1\in I^X$ such that for $\mathit{\alpha }=\frac{1}{2}$ and $\mathit{\beta }=\frac{1}{2}$ in $I_0$, we get that $f=x_1$ in ${\mathcal{N}}_{\{x\}}^{\mathit{\alpha }}$ and $g=y_1$ in ${\mathcal{N}}_{\{y\}}^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })=\frac{1}{2}>\sup (x_1\wedge y_1)=0$ and thus, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })$ normal space. Also, it is an $(\mathit{\alpha },\mathit{\beta })$$T_1$-space. Hence, $(X,\mathit{\tau })$ is an $(\mathit{\alpha },\mathit{\beta })$$T_4$-space

Proposition 5.11

A topological space (X, T) is $T_3$ if and only if the induced fuzzy topological space $(X,\mathit{\omega }(T))$ is $(\mathit{\alpha },\mathit{\beta })T_3$.

Proof

(X, T) is $T_1$ iff $(X,\mathit{\omega }(T))$ is $(\mathit{\alpha },\mathit{\beta })T_1$ is proved in Proposition 4.2.

Let $F={\mathrm{cl}}_{\mathit{\tau }}F$ and $x\notin F$ in X. Then, there are ${\mathcal{O}}_x,{\mathcal{O}}_F\in T$ such that ${\mathcal{O}}_x\cap {\mathcal{O}}_F=\mathrm{\varnothing }$. Taking $f={\mathit{\chi }}_{F^c}\text{and}g={\mathit{\chi }}_{{\mathcal{O}}_F}$ in $\mathit{\omega }(T)$, we get that ${\mathrm{int}}_{\mathit{\omega }(T)}f(x)\wedge {\mathrm{int}}_{\mathit{\omega }(T)}g(F)=1>0=\sup (f\wedge g)$. Hence, there are ${\mathcal{N}}_x^1$ and ${\mathcal{N}}_F^1$ of x and F respectively, such that $1>\sup (f\wedge g)$ and thus, $(X,\mathit{\omega }(T))$ is an $(\mathit{\alpha },\mathit{\beta })T_3$-space.

Conversely, $F={\mathrm{cl}}_{\mathit{\tau }}F$ and $x\notin F$ imply there are $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_y^{\mathit{\beta }}$ for all $y\in F$ such that ${\mathrm{int}}_{\mathit{\omega }(T)}f(x)\wedge {\mathrm{int}}_{\mathit{\omega }(T)}g(F)\ge (\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha }\wedge \mathit{\beta }\in I_0$. That is, ${\mathrm{int}}_{\mathit{\omega }(T)}f\in \mathit{\omega }(T),x\in s_{\mathit{\alpha }}{\mathrm{int}}_{\mathit{\omega }(T)}f$ and ${\mathrm{int}}_{\mathit{\omega }(T)}g\in \mathit{\omega }(T),F\in s_{\mathit{\beta }}{\mathrm{int}}_{\mathit{\omega }(T)}g$, which means that $s_{\mathit{\alpha }}{\mathrm{int}}_{\mathit{\omega }(T)}f={\mathcal{O}}_x\in T$, $s_{\mathit{\beta }}{\mathrm{int}}_{\mathit{\omega }(T)}g={\mathcal{O}}_F\in T$ and moreover, ${\mathcal{O}}_x\cap {\mathcal{O}}_F=\mathrm{\varnothing }$, and thus, (X, T) is a $T_3$-space. $\square$

Proposition 5.12

A topological space (X, T) is $T_4$ iff the induced fuzzy topological space $(X,\mathit{\omega }(T))$ is an $(\mathit{\alpha },\mathit{\beta })T_4$.

Proof

Similar to Proposition 6.3. $\square$

Proposition 6.5

Let $(X,\mathit{\tau })$ be an $(\mathit{\alpha },\mathit{\beta })$$T_3$-space, and let $\mathit{\sigma }$ be an fuzzy topology on X finer than $\mathit{\tau }$. Then, $(X,\mathit{\sigma })$ is also an $(\mathit{\alpha },\mathit{\beta })$$T_3$-space.

Proof

Let $x\in X$ and F be a closed subset of X with $x\notin F$. Then, there are $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_F^{\mathit{\beta }}$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$, that is $\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\tau }}f(x)$, $\mathit{\beta }\le {\mathrm{int}}_{\mathit{\tau }}g(y)$ for all $y\in F$ and $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$, which means that $\mathit{\alpha }\le {\mathrm{int}}_{\mathit{\sigma }}f(x)$, $\mathit{\beta }\le {\mathrm{int}}_{\mathit{\sigma }}g(y)$ for all $y\in F$ and $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$ and thus, $f\in {\mathcal{N}}_x^{\mathit{\alpha }}$ and $g\in {\mathcal{N}}_F^{\mathit{\beta }}$ in $(X,\mathit{\sigma })$ such that $(\mathit{\alpha }\wedge \mathit{\beta })>\sup (f\wedge g)$; $\mathit{\alpha },\mathit{\beta }\in I_0$. Hence, $(X,\mathit{\sigma })$ is an $(\mathit{\alpha },\mathit{\beta })$ regular space. Proposition 4.3 states that $(X,\mathit{\sigma })$ is an $(\mathit{\alpha },\mathit{\beta })$$T_1$-space, and thus, it is an $(\mathit{\alpha },\mathit{\beta })$$T_3$-space. $\square$

Proposition 6.6

Let $(X,\mathit{\tau })$ be an $(\mathit{\alpha },\mathit{\beta })$$T_4$-space and let $\mathit{\sigma }$ be a fuzzy topology on X finer than $\mathit{\tau }$. Then, $(X,\mathit{\sigma })$ is also an $(\mathit{\alpha },\mathit{\beta })$$T_4$-space.

Proof

Similar to the proof in Proposition 6.5. $\square$

Additional information

Funding

The author received no direct funding for this research.