Fuzzy simple expansion

Let $(X\text{,}\tau )$ be a fuzzy topological space, $\eta \in I^X$ such that $\eta \notin \tau$ . Then the fuzzy topology ${\tau }_{\eta }=\{\rho \vee ({\rho }^{\prime }\wedge \eta ):\rho \text{,}{\rho }^{\prime }\in \tau \}$ is called a simple expansion of $\tau$ by $\eta$ .

Let $$a(x)=\left\{\begin{array}{cc}x\text{,}\hfill & \mathrm{if}0⩽x⩽\frac{1}{2}\hfill \\ 1-x\text{,}\hfill & \mathrm{if}\frac{1}{2}⩽x⩽1\hfill \end{array}\right.\text{,}b(x)=\left\{\begin{array}{cc}1-2x\text{,}\hfill & \mathrm{if}0⩽x⩽\frac{1}{2}\hfill \\ \frac{1}{2}\text{,}\hfill & \mathrm{if}\frac{1}{2}⩽x⩽1\hfill \end{array}\right.\text{,}$$ $$c(x)=\left\{\begin{array}{cc}\frac{3}{2}x\text{,}\hfill & \mathrm{if}0⩽x⩽\frac{1}{2}\hfill \\ \frac{3}{4}\text{,}\hfill & \mathrm{if}\frac{1}{2}⩽x⩽1\hfill \end{array}\right.\text{,}d(x)=\left\{\begin{array}{cc}4x\text{,}\hfill & \mathrm{if}0⩽x⩽\frac{1}{4}\hfill \\ \frac{4}{3}(1-x)\text{,}\hfill & \mathrm{if}\frac{1}{4}⩽x⩽1\hfill \end{array}\right.\text{.}$$ Let $\tau =\{0_X\text{,}{1}_X\text{,}a\}$ be a fuzzy topology on X. Then ${\tau }_b=\{0_X\text{,}{1}_X\text{,}a\text{,}b\text{,}a\wedge b\text{,}a\vee b\}$ , ${\tau }_c=\{0_X\text{,}{1}_X\text{,}a\text{,}c\}$ and ${\tau }_d=\{0_X\text{,}{1}_X\text{,}a\text{,}d\}$ be three expansions for a fuzzy topology $\tau$ by using the fuzzy sets $b\text{,}c$ and d, respectively.

If $(X\text{,}\tau )$ is a fuzzy topological space and $\eta \in I^X$ , $\eta \notin \tau$ , then the topology ${\tau }_{\eta }=\{\rho \vee ({\rho }^{\prime }\wedge \eta ):\rho \text{,}{\rho }^{\prime }\in \tau \}$ coincides with the fuzzy topology ${\tau }^{*}$ generated by the class $\tau \cup \eta$ .

Let $\zeta \in {\tau }^{*}$ . Then $\zeta \in {\bigvee }_{i\in △}({\bigwedge }_{j\in {△}_0}{\lambda }_{\mathit{ij}})$ , ${\lambda }_{\mathit{ij}}\in \tau \cup \{\eta \}$ and $△({△}_0)$ is an arbitrary (finite) index set. If ${\lambda }_{\mathit{ij}}\in \tau$ for each $i\in △\text{,}j\in {△}_0$ , then $\zeta \in \tau \subset {\tau }_{\eta }$ . Assume that there exist $i_0\in △\text{,}{j}_0\in {△}_0$ such that ${\lambda }_{i_0{j}_0}=\eta \wedge \rho$ for some $\rho \in \tau$ . Therefore $\zeta =[{\bigvee }_{i\in △-\{i_0\}}({\bigwedge }_{j\in {△}_0-\{j_0\}}{\lambda }_{\mathit{ij}})]\vee (\eta \wedge \rho )$ , clearly this implies that $\zeta ={\rho }^{\prime }\vee (\rho \wedge \eta )$ , $\rho \text{,}{\rho }^{\prime }\in \tau$ . Hence $\zeta \in {\tau }_{\eta }$ .

Conversely, if $\zeta \in {\tau }_{\eta }$ , then there exist $\rho \text{,}{\rho }^{\prime }\in \tau$ such that $\zeta ={\rho }^{\prime }\vee (\rho \wedge \eta )$ . Therefore $\zeta \in {\tau }^{*}$ . □

Definition 2.2 Singal and Prakash, 1991

Let $\mu$ be any fuzzy set of a fts $(X\text{,}\tau )$

• $\mu$ is called fuzzy preopen set of X if $\mu ⩽\mathit{int}(\mathit{cl}(\mu ))$ .

• $\mu$ is called fuzzy preclosed set of X if $\mu ⩾\mathit{cl}(\mathit{int}(\mu ))$ .

Definition 2.3 Singal and Prakash, 1991

Fuzzy preclosure and fuzzy preinterior of a fuzzy set $\mu$ of a fts $(X\text{,}\tau )$ are defined as follows:

• $\mathit{pcl}(\mu )=\bigwedge \{\rho :\rho$ is a fuzzy preclosed and $\mu ⩽\rho \}$ .

• $\mathit{pint}(\mu )=\bigvee \{\nu :\nu$ is a fuzzy preopen and $\mu ⩾\nu \}$ .

In this paper we introduce new classes of fuzzy sets called fuzzy $\eta$ -preopen sets. This family of fuzzy sets not only depends on the fuzzy topology $\tau$ but also on ${\tau }_{\eta }$ . We study some of its properties and its characterizations.

For any fts $(X\text{,}\tau )$ . A fuzzy subset $\mu$ is called a fuzzy $\eta$ -preopen if $\mu ⩽{\mathit{int}}_{{\tau }_{\eta }}(\mathit{cl}(\mu ))$ .

The complement of fuzzy $\eta$ -preopen set is called fuzzy $\eta$ -preclosed set. The family of all fuzzy $\eta$ -preopen sets is denoted by ${\mathit{FP}}_{\eta }O(X)$ and the family of all fuzzy $\eta$ -preclosed sets is denoted by ${\mathit{FP}}_{\eta }C(X)$ .

The following example show that a fuzzy preopen set is a fuzzy $\eta$ -preopen but not conversely.

Consider the following fuzzy sets $$a(x)=\left\{\begin{array}{cc}x-\frac{1}{4}\text{,}\hfill & \mathrm{if}\frac{1}{4}⩽x⩽\frac{1}{2}\hfill \\ 0\text{,}\hfill & \mathrm{otherwise}\hfill \\ \frac{3}{4}-x\text{,}\hfill & \mathrm{if}\frac{1}{2}⩽x⩽\frac{3}{4}\hfill \end{array}\right.\text{,}b(x)=\left\{\begin{array}{cc}x\text{,}\hfill & \mathrm{if}0⩽x⩽\frac{1}{4}\hfill \\ \frac{1}{4}\text{,}\hfill & \mathrm{if}\frac{1}{4}⩽x⩽1\hfill \end{array}\right.\text{,}$$ $$c(x)=\left\{\begin{array}{cc}\frac{3}{2}x\text{,}\hfill & \mathrm{if}0⩽x⩽\frac{1}{2}\hfill \\ \frac{3}{4}\text{,}\hfill & \mathrm{if}\frac{1}{2}⩽x⩽1\hfill \end{array}\right.\text{,}\eta (x)=\left\{\begin{array}{cc}3x\text{,}\hfill & \mathrm{if}0⩽x⩽\frac{1}{4}\hfill \\ \frac{3}{4}\text{,}\hfill & \mathrm{if}\frac{1}{4}⩽x⩽1\hfill \end{array}\right.\text{.}$$ Let $\tau =\{0_X\text{,}{1}_X\text{,}a\text{,}b\}$ , $\eta \notin \tau$ . Then ${\tau }_{\eta }=\{0_X\text{,}{1}_X\text{,}a\text{,}b\text{,}\eta \}$ . We have the fuzzy set c is a fuzzy $\eta$ -preopen but it is not fuzzy preopen, since $\mathit{int}(\mathit{cl}(c))=b\ngeqq c$ .

Let $(X\text{,}\tau )$ be a fts, $\eta$ be a fuzzy set such that $\eta \notin \tau$ . We have:

• (1)

  An arbitrary union of fuzzy $\eta$ -preopen sets is a fuzzy $\eta$ -preopen set.

• (2)

  An arbitrary intersection of fuzzy $\eta$ -preclosed sets is a fuzzy $\eta$ -preclosed set.

(1) Let $\{V_{\alpha }\}$ be a collection of fuzzy $\eta$ -preopen sets. Then, for each $\alpha$ , $V_{\alpha }⩽{\mathit{int}}_{{\tau }_{\eta }}(\mathit{cl}(V_{\alpha }))$ . Now, $$\bigvee V_{\alpha }⩽\bigvee {\mathit{int}}_{{\tau }_{\eta }}(\mathit{cl}(V_{\alpha }))⩽{\mathit{int}}_{{\tau }_{\eta }}\bigvee (\mathit{cl}(V_{\alpha }))={\mathit{int}}_{{\tau }_{\eta }}(\mathit{cl}(\bigvee V_{\alpha }))$$ Hence $\bigvee V_{\alpha }$ is a fuzzy $\eta$ -preopen set.

(2) Follows easily by taking complements. □

Let $(X\text{,}\tau )$ be a fts and $\eta$ be a fuzzy set on X, $\eta \notin \tau$ . Fuzzy $\eta$ -preclosure $(p_{\eta }\mathit{cl})$ and fuzzy $\eta$ -preinterior $(p_{\eta }\mathit{int})$ of a fuzzy set $\varrho$ are defined as follows:

• $p_{\eta }\mathit{cl}(\varrho )=\bigwedge \{\mu :\mu \in {\mathit{FP}}_{\eta }C(X\text{,}\tau )\mathrm{and}\varrho ⩽\mu \}$ .

• $p_{\eta }\mathit{int}(\varrho )=\bigvee \{\nu :\nu \in {\mathit{FP}}_{\eta }O(X\text{,}\tau )\mathrm{and}\varrho ⩾\nu \}$ .

Let $\varrho$ be any fuzzy set in a fts $(X\text{,}\tau )$ , $\eta$ be a fuzzy set such that $\eta \notin \tau$ . Then $p_{\eta }-\mathit{cl}(1-\varrho )=1-p_{\eta }-\mathit{int}(\varrho )$ and $p_{\eta }-\mathit{int}(1-\varrho )=1-p_{\eta }-\mathit{cl}(\varrho )$ .

We see that a fuzzy $\eta$ -preopen set $\mu ⩽\varrho$ is precisely the complement of fuzzy $\eta$ -preopen set $\nu ⩾1-\varrho$ . Thus $$p_{\eta }\mathit{int}(\varrho )=\bigvee \{1-\nu :\nu \in {\mathit{FP}}_{\eta }C(X\text{,}\tau )\mathrm{and}\nu ⩾1-\varrho \}=1-\bigwedge \{\nu :\nu \in {\mathit{FP}}_{\eta }C(X\text{,}\tau )\mathrm{and}\nu ⩾1-\varrho \}=1-p_{\eta }\mathit{cl}(1-\varrho )$$ Similarly, $p_{\eta }\mathit{cl}(\varrho )=1-p_{\eta }\mathit{int}(1-\varrho )$ . □

For a fts $(X\text{,}\tau )$ , $\eta$ be a fuzzy set such that $\eta \notin \tau$ , then $\mu$ is called a fuzzy $\eta$ -preclosed (resp. fuzzy $\eta$ -preopen) if and only if $\mu =p_{\eta }\mathit{cl}(\mu )$ (resp. $\mu =p_{\eta }\mathit{int}(\mu )$ ).

Let $\mu \in {\mathit{FP}}_{\eta }C(X\text{,}\tau )$ . Since $\mu ⩽\mu$ , $\mu \in \{f:f\in {\mathit{FP}}_{\eta }C(X\text{,}\tau )\mathrm{and}\mu ⩽f\}\text{,}$ and $\mu ⩽f$ implies that $\mu =\bigwedge \{f:f\in {\mathit{FP}}_{\eta }C(X\text{,}\tau )\mathrm{and}\mu ⩽f\}$ , i.e., $\mu =p_{\eta }\mathit{cl}(\mu )$ . Conversely, suppose that $\mu =p_{\eta }\mathit{cl}(\mu )$ , i.e., $\mu =\bigwedge \{f:f\in {\mathit{FP}}_{\eta }C(X\text{,}\tau )\mathrm{and}\mu ⩽f\}$ . This implies that $\mu \in \{f:f\in {\mathit{FP}}_{\eta }C(X\text{,}\tau )\mathrm{and}\mu ⩽f\}$ . Hence $\mu \in {\mathit{FP}}_{\eta }C(X\text{,}\tau )$ . □

For a fts $(X\text{,}\tau )$ , $\eta$ is a fuzzy set such that $\eta \notin \tau$ , the following hold for $\eta$ -preclosure

• (1)

  $p_{\eta }\mathit{cl}(O_X)=O_X$ .

• (2)

  $p_{\eta }\mathit{cl}(\mu )\in {\mathit{FP}}_{\eta }C(X\text{,}\tau )$ .

• (3)

  $p_{\eta }\mathit{cl}(\mu )⩽p_{\eta }\mathit{cl}(\nu )$ if $\mu ⩽\nu$ .

• (4)

  $p_{\eta }\mathit{cl}(p_{\eta }\mathit{cl}(\mu ))=p_{\eta }\mathit{cl}(\mu )$ .

Obvious. □

For a fts $(X\text{,}\tau )$ , $\eta$ be a fuzzy set such that $\eta \notin \tau$ , the following are hold

• (1)

  $p_{\eta }\mathit{cl}(\mu \vee \nu )⩾p_{\eta }\mathit{cl}(\mu )\vee p_{\eta }\mathit{cl}(\nu )$ .

• (2)

  $p_{\eta }\mathit{cl}(\mu \wedge \nu )⩽p_{\eta }\mathit{cl}(\mu )\wedge p_{\eta }\mathit{cl}(\nu )$ .

Obvious. □

Let $(X\text{,}\tau )$ and $(Y\text{,}\sigma )$ be two fts's. A function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is called:

• Fuzzy preopen (Bin Shahna, 1991) if $f(\lambda )$ is fuzzy preopen subset of Y, for each fuzzy open set $\lambda$ in X.

• Fuzzy open (Chang, 1968) if $f(\lambda )$ is fuzzy open subset of Y, for each fuzzy open set $\lambda$ in X.

• Fuzzy weakly open (Park et al., 1997) if $f(\lambda )⩽(\mathit{int}(f(\mathit{cl}(\lambda ))))$ , for each fuzzy open set $\lambda$ in X.

• Fuzzy contra open (Caldas et al., 2004) (resp. fuzzy contra closed) if $f(\lambda )$ is fuzzy closed (resp. fuzzy open) set of Y, for each fuzzy open (resp. fuzzy closed) set $\lambda$ in X.

• Fuzzy weakly preopen (Caldas et al., 2004) if $f(\lambda )⩽(\mathit{pint}(f(\mathit{cl}(\lambda ))))$ , for each fuzzy open set $\lambda$ in X.

Let $(X\text{,}\tau )$ and $(Y\text{,}\sigma )$ be two fts's, $\eta$ be a fuzzy set such that $\eta \notin \sigma$ . A function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is called:

• Fuzzy $\eta$ -preopen if $f(\lambda )$ is a fuzzy $\eta$ -preopen subset of Y, for each fuzzy open set $\lambda$ in X.

• Fuzzy $\eta$ -open if $f(\lambda )$ is fuzzy $\eta$ -open subset of Y, for each fuzzy open set $\lambda$ in X.

• Fuzzy weakly $\eta$ -open if $f(\lambda )⩽({\sigma }_{\eta }\mathit{int}(f(\mathit{cl}(\lambda ))))$ , for each fuzzy open set $\lambda$ in X.

• Fuzzy weakly $\eta$ -preopen if $f(\lambda )⩽(p_{\eta }\mathit{int}(f(\mathit{cl}(\lambda ))))$ , for each fuzzy open set $\lambda$ in X.

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Let $(X\text{,}\tau )$ and $(Y\text{,}\sigma )$ be two fts's, $\eta$ be a fuzzy set such that $\eta \notin \sigma$ . For a function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ the following are equivalent.

• (1)

  f is a fuzzy weakly $\eta$ -preopen.

• (2)

  $f(\mathit{int}(\mu ))⩽p_{\eta }\mathit{int}(f(\mu ))\forall \mu \in \mathit{FC}(X)$ .

• (3)

  $f(\mathit{int}(\mathit{cl}(\lambda )))⩽p_{\eta }\mathit{int}(f(\mathit{cl}(\lambda )))\forall \lambda \in \mathit{FO}(X)$ .

• (4)

  $f(\lambda )⩽p_{\eta }\mathit{int}(f(\mathit{cl}(\lambda )))\forall \lambda \in \mathit{FPO}(X)$ .

Obvious. □

Definition 3.2 Caldas et al., 2004

A function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is said to be fuzzy strongly continuous, if for every fuzzy subset $\lambda$ of X, $f(\mathit{cl}(\lambda ))⩽f(\lambda )$ .

Let $(X\text{,}\tau )$ and $(Y\text{,}\sigma )$ be two fts's, $\eta$ be a fuzzy set such that $\eta \notin \sigma$ . If a function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is a fuzzy weakly $\eta$ -preopen and fuzzy strongly continuous. Then it is a fuzzy $\eta$ -preopen.

Let $\lambda$ be a fuzzy subset of X. Since f is fuzzy weakly $\eta$ -preopen, then we have $f(\lambda )⩽p_{\eta }\mathit{int}(f(\mathit{cl}(\lambda )))$ . However, because f is strongly continuous, $f(\lambda )⩽p_{\eta }\mathit{int}(f(\lambda ))$ , i.e., $f(\lambda )=p_{\eta }\mathit{int}(f(\lambda ))$ . Consequently, $f(\lambda )\in {\mathit{FP}}_{\eta }O(Y)$ . Therefore f is fuzzy $\eta$ -preopen function. □

Let $(X\text{,}\tau )$ be a fuzzy regular space and $(Y\text{,}\sigma )$ be a fts, $\eta$ be a fuzzy set such that $\eta \notin \sigma$ . Then the function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is a fuzzy weakly $\eta$ -preopen if and only if f is a fuzzy $\eta$ -preopen.

The sufficiency is clear. For the necessity, let $\lambda$ be a non-null fuzzy open subset of X. For each $x_p$ fuzzy point in $\lambda$ , let ${\mu }_{x_p}$ be a fuzzy open set such that $x_p\in {\mu }_{x_p}⩽\mathit{cl}({\mu }_{x_p})⩽\lambda$ . Hence we obtain that $\lambda =\bigcup \{{\mu }_{x_p}:x_p\in \lambda \}=\bigcup \{\mathit{cl}({\mu }_{x_p}):x_p\in \lambda \}$ and $$f(\lambda )=\bigcup \{f({\mu }_{x_p}):x_p\in \lambda \}⩽\bigcup \{p_{\eta }\mathit{int}(f(\mathit{cl}({\mu }_{x_p}))):x_p\in \lambda \}⩽p_{\eta }\mathit{int}(f(\bigcup \{\mathit{cl}({\mu }_{x_p}))):x_p\in \lambda \}))=p_{\eta }\mathit{int}(f(\lambda ))\text{.}$$ Thus f is a fuzzy $\eta$ -preopen function. □

Definition 3.3 Pu and Liu, 1980

An fuzzy set $\mu$ is said be quasi-coincident (q-coincident) with an fuzzy set $\lambda$ , if there exists at least one point $x\in X$ such that $\mu (x)+\lambda (x)>1$ . It is denoted by $\mu q\lambda$ . $\mu \overline{q}\lambda$ means that $\mu$ and $\lambda$ are not q-coincident. For two fuzzy $\mu$ and $\lambda$ . $\mu ⩽\lambda$ iff $\mu (x)⩽\lambda (x)$ for each $x\in X$ . Note that $\mu ⩽\lambda$ iff $\mu \overline{q}X\backslash \lambda$ .

Definition 3.4 Caldas et al., 2004

Two non-empty fuzzy sets $\lambda$ and $\beta$ in a fuzzy topological space $(X\text{,}\tau )$ (i.e., neither $\lambda$ nor $\beta$ is $0_X$ ) are said to be fuzzy preseparated if $\lambda \overline{q}\mathit{pcl}(\beta )$ and $\beta \overline{q}\mathit{pcl}(\lambda )$ or equivalently if there exist two fuzzy preopen sets $\mu$ and $\nu$ such that $\lambda ⩽\mu$ , $\beta ⩽\nu$ , $\lambda \overline{q}\nu$ and $\beta \overline{q}\mu$ .

Two non-empty fuzzy sets $\lambda$ and $\beta$ in a fuzzy topological space $(X\text{,}\tau )$ (i.e., neither $\lambda$ nor $\beta$ is $0_X$ ) are said to be fuzzy $\eta$ -preseparated if $\lambda \overline{q}{p}_{\eta }\mathit{cl}(\beta )$ and $\beta \overline{q}{p}_{\eta }\mathit{cl}(\lambda )$ or equivalents if there exist two fuzzy $\eta$ -preopen sets $\mu$ and $\nu$ such that $\lambda ⩽\mu$ , $\beta ⩽\nu$ , $\lambda \overline{q}\nu$ and $\beta \overline{q}\mu$ .

A fuzzy topological space which cannot be expressed as the union of two fuzzy preseparated (fuzzy $\eta$ -preseparated) sets is said to be a fuzzy preconnected (fuzzy $\eta$ -preconnected).

Let $(X\text{,}\tau )$ and $(Y\text{,}\sigma )$ are the fuzzy topological spaces, $\eta$ be a fuzzy set such that $\eta \notin \sigma$ . If $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is an injective fuzzy weakly $\eta$ -preopen function of a space X onto a fuzzy $\eta$ -preconnected space Y, then X is fuzzy connected.

Let X be not fuzzy connected. Then there exist fuzzy separated sets $\beta$ and $\gamma$ in X, such that $X=\beta \cup \gamma$ . Since $\beta$ and $\gamma$ are fuzzy separated. There exists two fuzzy open sets $\mu$ and $\nu$ such that $\beta ⩽\mu$ , $\gamma ⩽\nu$ , $\beta \overline{q}\nu$ and $\gamma \overline{q}\mu$ . Hence we have $f(\beta )⩽f(\mu )$ , $f(\gamma )⩽f(\nu )$ , $f(\beta )\overline{q}f(\nu )$ and $f(\gamma )\overline{q}f(\mu )$ . Since f is fuzzy weakly $\eta$ -preopen, we have $f(\mu )⩽p_{\eta }\mathit{int}(f(\mathit{cl}(\mu )))$ and $f(\nu )⩽p_{\eta }\mathit{int}(f(\mathit{cl}(\nu )))$ and since $\mu$ and $\nu$ are fuzzy open and also fuzzy closed, we have $f(\mathit{cl}(\mu ))=f(\mu )$ , $f(\mathit{cl}(\nu ))=f(\nu )$ . Hence $f(\mu )$ and $f(\nu )$ are fuzzy $\eta$ -preopen in Y. Therefore, $f(\beta )$ and $f(\gamma )$ are fuzzy $\eta$ -preseparated sets in Y and $Y=f(X)=f(\beta \cup \gamma )=f(\beta )\cup f(\gamma )$ . Hence this contrary to the fact that Y is fuzzy $\eta$ -preconnected. Thus X is fuzzy connected. □

Definition 3.6 Caldas et al., 2004

Space X is said to be fuzzy hyper connected if every non-null fuzzy open subset of X is fuzzy dense in X.

Let $(X\text{,}\tau )$ and $(Y\text{,}\sigma )$ be two fts's, $\eta$ be a fuzzy set such that $\eta \notin \sigma$ . If X is a fuzzy hyper connected space. Then $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is fuzzy weakly $\eta$ -preopen if and only if $f(X)$ is fuzzy $\eta$ -preopen set in Y.

The sufficiency is trivial. For the necessity observe that for any fuzzy open subset $\lambda$ os X, $f(\lambda )⩽f(X)=p_{\eta }\mathit{int}(f(X))=p_{\eta }\mathit{int}(f(\mathit{cl}(\lambda )))$ . □

A function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is said to be fuzzy weakly $\eta$ -preclosed if $p_{\eta }\mathit{cl}(f(\mathit{int}(\beta )))⩽f(\beta )$ for each fuzzy closed subset $\beta$ of X.

Let $(X\text{,}\tau )$ and $(Y\text{,}\sigma )$ be two fts's, $\eta$ be a fuzzy set such that $\eta \notin \sigma$ . A function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is called:

• Fuzzy $\eta$ -preclosed if $f(\lambda )$ is a fuzzy $\eta$ -preclosed subset of Y, for each fuzzy closed set $\lambda$ in X.

• Fuzzy $\eta$ -closed if $f(\lambda )$ is fuzzy $\eta$ -closed subset of Y, for each fuzzy closed set $\lambda$ in X.

• Fuzzy weakly $\eta$ -closed if $({\sigma }_{\eta }\mathit{cl}(f(\mathit{int}(\lambda ))))⩽f(\lambda )$ , for each fuzzy open set $\lambda$ in X.

• Fuzzy weakly $\eta$ -preclosed if $(p_{\eta }\mathit{cl}(f(\mathit{int}(\lambda ))))⩽f(\lambda )$ , for each fuzzy open set $\lambda$ in X.

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For a function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ , $\eta \notin \sigma$ , the following conditions are equivalent:

• (1)

  f is fuzzy weakly $\eta$ -preclosed.

• (2)

  $p_{\eta }\mathit{cl}(f(\lambda ))⩽f(\mathit{cl}(\lambda ))$ for every open set $\lambda$ in X.

$(1)\Rightarrow (2)$ Let $\lambda$ be any fuzzy open subset of X. Then $$p_{\eta }\mathit{cl}(f(\lambda ))=p_{\eta }\mathit{cl}(f(\mathit{int}(\lambda )))⩽p_{\eta }\mathit{cl}(f(\mathit{int}(\mathit{cl}(\lambda ))))⩽f(\mathit{cl}(\lambda ))\text{.}$$

$(2)\Rightarrow (1)$ Let $\beta$ be any fuzzy closed subset of X. Then $$p_{\eta }\mathit{cl}(f(\mathit{int}(\beta )))⩽f(\mathit{cl}(\mathit{int}(\beta )))⩽f(\mathit{cl}(\beta ))=f(\beta ).\square$$

For a function $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ , $\eta \notin \sigma$ , the following conditions are equivalent:

• (1)

  f is fuzzy weakly $\eta$ -preclosed.

• (2)

  $p_{\eta }\mathit{cl}(f(\beta ))⩽f(\mathit{cl}(\beta ))$ for each fuzzy open subset $\beta$ of X.

• (3)

  $p_{\eta }\mathit{cl}(f(\mathit{int}(\beta )))⩽f(\beta )$ for each fuzzy closed subset $\beta$ of X.

Obvious. □

Let $(X\text{,}\tau )$ and $(Y\text{,}\sigma )$ are two fts's, $\eta \notin \sigma$ . Then:

• (1)

  If $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is fuzzy preclosed and fuzzy contra closed, then f is fuzzy weakly $\eta$ -preclosed.

• (2)

  If $f:(X\text{,}\tau )\to (Y\text{,}\sigma )$ is fuzzy contra open, then f is fuzzy weakly $\eta$ -preclosed.

(1) Let $\beta$ be a fuzzy closed subset of X. Since f is a fuzzy preclosed, then $\mathit{cl}(\mathit{int}(f(\beta )))⩽f(\beta )$ and since f is fuzzy contra closed, $f(\beta )$ is fuzzy open. Therefore $p_{\eta }\mathit{cl}(f(\mathit{int}(\beta )))⩽p_{\eta }\mathit{cl}(f(\beta ))⩽\mathit{cl}(\mathit{int}(f(\beta )))⩽f(\beta )$ . Consequently f is a fuzzy weakly $\eta$ -preclosed.

(2) Let $\beta$ be a fuzzy closed subset of X. Then $p_{\eta }\mathit{cl}(f(\mathit{int}(\beta )))⩽(f(\mathit{int}(\beta )))⩽f(\beta )$ . □