I am reading the paper Fuzzy Topological Spaces and Fuzzy Compactness by Robert Lowen. Lowen didn't wrote down his proof about proposition 3.1 since he thought it is trivial. But I would like to ask some help for the verification for my proof. Given some definition:
Let $\mathscr{J}(X)$ be the family of all topologies on $X$ and $\mathscr{W}(X)$ be the set of all fuzzy topologies on $X$. On $\mathbb{R}$, we consider the topology $\mathscr{J}_r=\{(\alpha,\infty) \cup \{\emptyset\} \vert \alpha \in \mathbb{R}\}$. The topological space one obtains unit interval $I$ the induced topology on $\mathscr{J}_r$ is denoted as $I_r$. Then we define the following maps. For the sets $\mathscr{J}(X)$ and $\mathscr{W}(X)$, we define the mapping \begin{align*} \iota : \mathscr{W}(X) &\to \mathscr{J}(X)\\ \delta &\mapsto \iota(\delta) \;\;\;\;\;\forall \delta \in \mathscr{W}(X), \end{align*} where $\iota(\delta)$ is a initial topology on $X$ for the family of "function" $\delta$ and the topological space $I_r$. Then we define the mapping \begin{align*} \omega : \mathscr{J}(X) &\to \mathscr{W}(X)\\ \mathscr{T} &\mapsto \omega(\mathscr{T})\;\;\;\;\;\forall \mathscr{T} \in \mathscr{J}(X), \end{align*} where $\omega(\mathscr{J})=\mathscr{G}(\mathscr{J},I_r)$ is a continuous function from $(X,\mathscr{J})$ to $I_r$. For every $\delta \in \mathscr{W}(X)$, $\delta$ is said to be $\textbf{topologically generated}$ if $\delta = \omega(\mathscr{T})$ for some $\mathscr{T} \in \mathscr{J}(X)$.
Let $f$ be a function from $X$ to $Y$. Let $B$ be a fuzzy set in $Y$ with membership function $\mu_{\scriptscriptstyle{B}}(x)$ for all $x$ in $X$. Then the inverse of $B$, denoted as $f^{-1}[B]$, is a fuzzy set in $X$ whose membership function is given by \begin{align*} \mu_{\scriptscriptstyle{f^{-1}[B]}}(x) = \mu_{\scriptscriptstyle{B}}(f(x)) \text{ for all } x \in X. \end{align*} Conversely, let $A$ be a fuzzy set in $X$ with membership function $\mu{\scriptscriptstyle{A}}(x)$ for all $x$ in $X$. Then, the image of $A$, denoted as $f[A]$, is a fuzzy set in $Y$ whose membership function is given by \begin{align*} \mu_{\scriptscriptstyle{f[A]}}(y)=\begin{cases} \displaystyle\sup_{\scriptscriptstyle{x \in f^{-1}(y)}}\{\mu{\scriptscriptstyle{A}}(x)\} \hspace{0.5cm}& \text{if } f^{-1}(y) \neq \emptyset,\\ 0 & \text{otherwise. } \end{cases} \end{align*} for all $x$ in $X$ and $ y$ in $Y$ where $f^{-1}(y) = \{x \vert f(x) = y\}$.
A function $\tilde{f}$ from a fuzzy topological space ($fts$) $U_1 = (X, \delta)$ to another $fts$ $U_2=(Y,\gamma)$ is $F$-continuous if and only if \begin{align*} \mu_{\scriptscriptstyle{ f^{-1}[\nu]}} \in \delta\;\;\; \forall \nu \in \gamma. \end{align*}
A function $\tilde{f} : (X,\delta) \to (Y,\gamma)$ is continuous if and only if $\tilde{f} : \big(X,\iota(\delta)\big) \to \big(Y,\iota(\gamma)\big)$ is continuous.
Proposition 3.1: Consider the following properties for $\tilde{f} : (X,\delta) \to (Y,\gamma)$:
- $\tilde{f}$ is $F$-continuous.
- $\tilde{f}$ is continuous.
- $\tilde{f} : (X,\bar{\delta}) \to (Y,\bar{\gamma})$ is $F$-continuous.
- $\tilde{f} : (X,\bar{\delta}) \to (Y,\gamma)$ is $F$-continuous.
then we have $(1) \Rightarrow (2) \Leftrightarrow (3) \Leftrightarrow (4)$.
For the case (1) $\Rightarrow$ (2):
Assume a basis for $\iota(\delta)$ and $\iota(\gamma)$ respectively are given by $\displaystyle\bigcap_{i=1}^n \mu_i^{-1}(r_i,1]$ and $\displaystyle\bigcap_{j=1}^n \nu_j^{-1} (r_j,1]$ where $\mu_i \in \delta$ and $\nu_j \in \gamma$. Recall that for $f : X \to Y$, \begin{align*} \mu_i (x) = f^{-1} \big(\nu_i(x)\big) = \nu_i\big(f(x)\big) \;\;\forall x \in X. \end{align*} Then if $\tilde{f}$ is $F$-continuous, we have \begin{align*} f^{-1} (\nu_i) = \nu_i\big(f(x)\big) = \mu_i(x) \in \delta\;\;\forall \nu_i \in \gamma. \end{align*} Recall that for each $y \in \nu_i^{-1}(r_j,1]$ for some $i$, we have $\nu_i(y) \in (r_i,1]$. Since $\nu_i^{-1}(r_i,1]$ is an open set in $\big(Y,\iota(\gamma)\big)$, we have \begin{align*} \mu_i (x) = f^{-1} (\nu_i) = \nu_i\big(f(x)\big) \in (r_i,1] \Rightarrow x \in \mu_i^{-1}(r_i,1]\;\; \forall x \in X \end{align*} where $\mu_i^{-1}(r_i,1]$ is an open set in $\big(X,\iota(\delta)\big)$. Then we have proved (1) $\Rightarrow$ (2).
For the case $(2) \Leftrightarrow (3)$:
Assume $\tilde{f}: (X,\delta) \to (Y,\gamma)$ is continuous. Since $\omega = \mathscr{G}(\mathscr{J},I_r)$, we have $\tilde{\mu} \in \mathscr{G}\big((X,\mathscr{T}),I_r\big)$ and $\tilde{\nu} \in \mathscr{G}\big((Y,\mathscr{P}),I_r\big)$ are both continuous. Since $\tilde{f}$ is continuous, then we have $\tilde{\mu}_i^{-1}(r_j,1]=f^{-1}\big(\tilde{\nu}_j^{-1}(r_j,1]\big) \in (X,\mathcal{J})$ for every $\tilde{\nu}_j(r_j,1] \in \bar{\gamma}$. Since $\tilde{\mu}_i \in \mathscr{G}\big((X,\mathscr{T}),I_r\big)$, then we have $\tilde{f} : (X,\bar{\delta}) \to (Y,\bar{\gamma})$ is $F$-continuous. Conversely, assume $\tilde{f} : (X,\bar{\delta}) \to (Y,\bar{\gamma})$ is $F$-continuous. Notice that $\omega\big(\iota(\gamma)\big) = \bar{\gamma}$. Since $\tilde{f} : (X,\bar{\delta}) \to (Y,\bar{\gamma})$ is $F$-continuous and $\bar{\delta}$ is the smallest topologically generated fuzzy topology that contains $\delta$, hence $\tilde{f}$ is continuous.
For the case $(3)\Leftrightarrow (4)$:
Assume $\tilde{f} : (X,\bar{\delta}) \to (Y,\bar{\gamma})$ is $F$-continuous. Then define a function $g$ as $g : Y \to Y$ and define $\tilde{g}:(Y,\bar{\gamma}) \to (Y,\gamma)$. Notice that $\bar{\gamma}$ is the smallest topologically generated fuzzy topology that contains $\gamma$, namely \begin{align*} \nu \in \bar{\gamma}\;\;,\forall \nu \in \gamma. \end{align*} Then $\tilde{g}$ is $F$-continuous. Since $\tilde{f}$ and $\tilde{g}$ are both $F$-continuous, then $\tilde{g} \circ \tilde{f} : (X,\bar{\delta}) \to (Y,\gamma)$ is $F$-continuous. Conversely, assume $\tilde{f} : (X,\bar{\delta}) \to (Y,\gamma)$ is $F$-continuous. Since $(1)\Rightarrow(2)$, then we have $\tilde{f} : \big(X,\iota(\bar{\delta})\big) \to \big(Y,\iota(\gamma)\big)$ is continuous. By $(2) \Rightarrow (3)$, we have $\tilde{f} : (X,\bar{\delta}) \to (Y,\bar{\gamma})$ is $F$-continuous.
Can someone help me to check my proof works or not? Thanks for your help.
