On Variations of Fuzzy Homogeneity

Abstract

Some of the important concepts in ordinary topological spaces like $n$ -homogeneity, strongly locally homogeneity, almost countable dense homogeneity, and transposition homogeneity are extended to fuzzy topological spaces and the relationship between these spaces together with fuzzy semi-discreteness is well studied here. Since there are spaces that fail to be fuzzy countable dense homogeneous ( $F C D H$ ) or fuzzy semi discrete ( $F S D$ ), only due to finitely many points, we found it necessary to study such spaces, which we refer to as almost fuzzy countable dense homogeneous ( $A F C D H$ ) and almost fuzzy semi discrete spaces ( $A F S D$ ) respectively. It is shown that the fuzzy extensions of the above notions are all fuzzy topological properties. A further variant of almost countable dense homogeneity called $A F C D H^{*}$ was introduced to fuzzy spaces and proved that a countable fuzzy separable space is $A F S D$ if and only if it is $A F C D H$ or $A F C D H^{*}$ . Hence, in countable fuzzy separable spaces, the notions $A F C D H$ and $A F C D H^{*}$ are equivalent. One of the main results identifies conditions under which an $A F C D H$ space becomes an $F C D H$ space. Further, the paper discusses ordinary topological spaces generated by fuzzy topological spaces and we shall observe that whenever the fuzzy topological space is almost fuzzy semi discrete, or fuzzy transposition homogeneous or almost fuzzy countable dense homogeneous, so is the corresponding ordinary space.

Keywords

fuzzy semi-discrete fuzzy strongly locally homogeneity almost fuzzy countable dense homogeneity almost fuzzy CDH of type *fuzzy transposition homogeneity

1. Introduction

Since L.A. Zadeh introduced fuzzy sets in 1965 (Zadeh, 1965), the field has grown over the past 60 years into an active area of research that connects traditional math with the study of uncertainty and imprecision. Some of the milestones in this integration include the development of fuzzy topologies by Chang (1968) and fuzzy metric spaces by Kramosil and Michálek (1975). Apart from its theoretical advancements, the practical application of fuzzy sets includes image processing (Selvy & Vinod Kumar, 2024), natural language processing, pattern recognition, decision-making, and optimization problems. Motivated by these developments, many authors have extended classical topological concepts to the fuzzy setting. In this paper, we extend different homogeneity concepts like $n$ -homogeneity (Ungar, 1978), almost countable dense homogeneity (Tallafha & Al-Rawashdeh, 2010), strongly locally homogeneity (Van Mill, 1982), and transposition homogeneity (Tallafha & Al-Rawashdeh, 2010) to fuzzy topological spaces and examine the relationships among these fuzzy analogues. The fuzzy extension of countable dense homogeneity by Al-Ghour and Fora (2016) laid the foundation for this paper. From the literature, we see two types of denseness for a collection of fuzzy points namely dense $(I)$ and dense $(I I)$ and, a type of homogeneity in ordinary space called almost $C D H$ . Based on these ideas, new concepts called almost fuzzy $C D H (I)$ and almost fuzzy $C D H (I I)$ were developed. However, the equivalence between these notions leads to the general concept of almost fuzzy CDH ( $A F C D H$ ) which generalizes the fuzzy $C D H$ space of Al-Ghour and Fora (2016). In particular, this generalization accounts for cases where two countable dense subsets $C$ and $D$ are not homeomorphic due to finitely many points; by appropriately adding points to $C$ and $D$ , homeomorphism can be achieved. Conversely, there exist spaces where removing finitely many points makes the countable dense subsets homeomorphic. Such spaces are called almost fuzzy $C D H$ of type *. Finally, this framework can be further extended to intuitionistic fuzzy sets, paving the way for broader applications in pattern recognition and robustness analysis. Intuitionistic fuzzy sets, introduced by Atanassov (1999), provide a more flexible representation of uncertainty through three parameters: membership, non-membership, and hesitation degrees. The concluding section of this paper discusses about ordinary topological spaces called $a$ -cut topological space $(Υ, T_{a})$ for $a \in [0, 1)$ that are generated by the fuzzy topological space $(Υ, T)$ and we shall observe that whenever the fuzzy topological space $(Υ, T)$ is almost fuzzy semi-discrete or fuzzy transposition homogeneous or almost fuzzy countable dense homogeneous (AFCDH), the corresponding ordinary space $(Υ, T_{0})$ also inherits these properties.

2. Preliminary Concepts

Some basic notations and definitions needed for our study are given here: Let $Υ$ be a non empty set. A fuzzy set $μ$ on $Υ$ is a function from $Υ$ to $[0, 1]$ (i.e., $μ \in [0, 1]^{Υ})$ . The fuzzy set that maps all elements to 0 (or 1 resp.) is denoted by 0 (or 1 resp.). If ${μ_{i} : i \in I}$ is a collection of fuzzy sets on $Υ$ , then $\forall x \in Υ$ , $(\cup μ_{i}) (x) = s u p {μ_{i} (x) : i \in I}$ and $(\cap μ_{i}) (x) = i n f {μ_{i} (x) : i \in I}$ . Let $x \in Υ$ and $0 < a < 1$ , then a fuzzy point $x_{a}$ on $Υ$ is the fuzzy set mapping only $x$ to $a$ and all other elements to $0$ . Here, $x \in Υ$ is called the support of $x_{a}$ . A fuzzy set $x_{a}$ is a fuzzy crisp point, denoted as $x_{I}$ if $x_{a} (y) = 1$ for $y = x$ and $x_{a} (y) = 0$ for $y \neq x$ . For any two fuzzy sets $λ$ and $μ$ , $λ$ is a subset of $μ$ denoted as $λ \subseteq μ$ iff $λ (x) \leq μ (x)$ $\forall x \in Υ$ and $λ = μ$ iff $λ \subseteq μ$ and $μ \subseteq λ$ . A fuzzy point $x_{a}$ is said to belong to a fuzzy set $μ$ , denoted as $x_{a} \in μ$ iff $x_{a} (x) < μ (x)$ , i.e., iff $a < λ (x)$ . Two fuzzy points are said to be distinct iff their supports are distinct. A family $T \subseteq [0, 1]^{Υ}$ of fuzzy sets is said to be a fuzzy topology for $Υ$ if it satisfies the three axioms: (i) $0, 1 \in T$ , (ii) $\forall$ $μ_{1}, μ_{2} \in T ⟹ μ_{1} \cap μ_{2} \in T$ and (iii) $\forall$ $(μ_{i}) \subseteq T ⟹ \cup μ_{i} \in T$ . The elements of $T$ are called as fuzzy open sets w.r.t $T$ . A fuzzy set is said to be closed w.r.t. $T$ iff it is equal to $1 - λ$ for some $λ \in T$ , and is denoted as $λ^{c}$ . The closure of a fuzzy set $λ$ is $μ$ iff $μ$ is the smallest (w.r.t the images) closed set such that $λ \subseteq μ$ , denoted as $C l (λ) = μ$ . A fuzzy topological space $(Υ, T)$ is said to be $T_{1}$ iff for any two distinct fuzzy points $p_{a}, q_{b}$ in $Υ$ , there exists $λ_{1}, λ_{2} \in T$ such that $p_{a} \in λ_{1}, λ_{1} \cap q_{b} = 0, λ_{2} \cap p_{a} = 0$ and $q_{b} \in λ_{2}$ . If $(Υ, T_{1})$ and $(Υ^{^{'}}, T_{2})$ are two fuzzy topological spaces and $h : (Υ, T_{1}) \to (Υ^{^{'}}, T_{2})$ is a bijective map, then $h$ is called a fuzzy homeomorphism iff both $h$ and $h^{- 1}$ are continuous. We denote $H (Υ, T)$ as the space of all fuzzy homeomorphisms on $Υ$ . Suppose $(Υ, T_{1})$ and $(Υ^{^{'}}, T_{2})$ are two fuzzy topological spaces such that $(Υ, T_{1})$ has the property $P$ and $h : (Υ, T_{1}) \to (Υ^{^{'}}, T_{2})$ is a fuzzy homeomorphism; then $P$ is said to be a fuzzy topological property if $(Υ^{^{'}}, T_{2})$ also satisfies $P$ . If $(Υ, T)$ is an ordinary topological space and $T_{u}$ is the usual Euclidean topology on $[0, 1]$ , then the class of all lower semi-continuous functions from $(Υ, T)$ to $([0, 1], T_{u})$ is a fuzzy topology on $Υ$ , denoted by $ω (T)$ . i.e. $ω (T) = {f : Υ \to [0, 1] ∣ f^{- 1} (a, 1] \in T,$ $\forall a \in [0, 1]}$ (Johnson, 1993).

Definition 2.1 Jose & Mathew, 2011

Let $Υ$ and $Υ^{'}$ be two sets, and let $h : Υ \to Υ^{'}$ be a function. Then, for any fuzzy set $g$ on $Υ$ , $h (g)$ is a fuzzy set on $Υ^{'}$ and for any fuzzy set $f$ on $Υ^{'}$ , $h^{- 1} (f)$ is a fuzzy set on $Υ$ , which are defined by

\begin{aligned} h (g) (y) & = {\begin{cases} s u p {g (x) ∣ x \in Υ, h (x) = y} & h^{- 1} (y) \neq \emptyset \\ 0 & h^{- 1} (y) = \emptyset \end{cases} \\ h^{- 1} (f) (x) = f (h (x)), \forall x \in Υ \end{aligned}

Remark 2.2
Therefore, if $h$ is a bijective function, then $h (g (y)) = g (h^{- 1} (y))$ , $\forall y \in Υ^{'}$ and $h^{- 1} (f) (x) = f (h (x))$ , $\forall x \in Υ$ .
Remark 2.3
If $h : Υ \to Υ$ is a bijective function and $x_{a}$ is a fuzzy point, then $h (x_{a}) = (h (x))_{a}$ . For,
$\begin{aligned} (h (x))_{a} (y) = {\begin{cases} a & y = h (x) \\ 0 & y \neq h (x) \end{cases} \\ h (x_{a}) (y) & = x_{a} (h^{- 1} (y)) = {\begin{cases} a & y = h (x) \\ 0 & y \neq h (x) \end{cases} \end{aligned}$

Definition 2.4 Al-Ghour & Fora, 2016

A fuzzy topological space $(Υ, T)$ is said to be fuzzy semi-discrete, abbreviated as $F S D$ iff for any $x \in Υ$ , the fuzzy crisp point $x_{I} \in T$ or there exists a fuzzy point $x_{a}$ for some $a$ such that $x_{a} \in T$ .

Clearly, $F S D$ is a fuzzy topological property.

Definition 2.5
Let $D$ be a collection of fuzzy points in the fuzzy topological space $(Υ, T)$ . The support of $D$ , denoted by $S (D)$ , is defined by
$S (D) = {x : x_{a} \in D}$

Lemma 2.6
Let $(Υ, T_{1})$ and $(Υ^{'}, T_{2})$ be two fuzzy topological spaces and $h : Υ \to Υ^{'}$ be a bijective map. Then $S (h (D)) = h (S (D))$ for any collection $D$ of fuzzy points of $(Υ, T)$ .
Proof.
The proof is straightforward. For instance, let $D = {x_{a}, y_{b}}$
$\begin{aligned} ⟹ & h (D) = {h (x_{a}), h (y_{b})} \\ ⟹ & h (D) = {(h (x))_{a}, (h (y))_{b}} \\ ⟹ & S (h (D)) = {h (x), h (y)} \\ ∴ S (D) = {x, y} ⟹ & h (S (D)) = {h (x), h (y)} \end{aligned}$

Definition 2.7
A fuzzy topological space $(Υ, T)$ is said to be fuzzy $n$ -homogeneous, if for any two sets $A$ and $B$ of $n$ distinct fuzzy points, there exists $h : (Υ, T) \to (Υ, T)$ such that
$h (S (A)) = S (B)$

For $n = 1$ , we simply say that the space is fuzzy homogeneous.
Theorem 2.8
Fuzzy $n$ -homogeneity is a fuzzy topological property.
Proof.
Let $(Υ, T_{1})$ and $(Υ^{'}, T_{2})$ be two fuzzy topological spaces such that $(Υ, T_{1})$ is fuzzy $n$ -homogeneous and $f : (Υ, T_{1}) \to (Υ^{'}, T_{2})$ be a homeomorphism. Suppose that $A$ and $B$ are two $n$ -element sets of distinct fuzzy points of $Υ^{'}$ . Then $f^{- 1} (A), f^{- 1} (B)$ are two $n$ -element sets of distinct fuzzy points of $Υ$ , and thus $\exists$ a homeomorphism $h : (Υ, T_{1}) \to (Υ^{'}, T_{2})$ such that $h (S (f^{- 1} (A))) = S (f^{- 1} (B))$ . By the above lemma, $h (f^{- 1} (S (A))) = f^{- 1} (S (B))$ or $(f \circ h \circ f^{- 1}) (S (A)) = S (B)$ .
Corollary 2.9
If $(Υ, T)$ is fuzzy homogeneous and $x_{a} or x_{I} \in T$ for some $x \in Υ$ , then $Υ$ is $F S D$ .
Proof.
Let $y \in Υ$ . Being fuzzy homogeneous $\exists$ a fuzzy homeomorphism $h$ on $Υ$ such that $h (x) = y$ . Clearly,
$\begin{aligned} h (x_{a}) (y) = x_{a} (x) = a and h (x_{a}) (z) = x_{a} (h^{- 1} (z)) = 0 \end{aligned}$
Since continuous image of an open set is open, $y_{a} = h (x_{a}) \in T$ and $y$ being arbitrary implies $Υ$ is $F S D$ .
Corollary 2.10
Suppose that $(Υ, T)$ is a fuzzy $n$ -homogeneous space and $A$ and $B$ are two sets of $n$ distinct fuzzy points. Define $H (A, B) = {h \in H (Υ, T) ∣ h (S (A)) = S (B)}$ . Then $H (A, B)$ is closed.
Proof.
Let $h \in \bar{H (A, B)}$ . Then $\exists$ $(h_{n}) \subseteq H (A, B)$ such that $h_{n} \to h$ and there by implying $h_{n} (x_{i}) \to h (x_{i})$ if $S (A) = {x_{1}, x_{2}, . ., x_{n}}$ . Since $\forall i$ , $(h_{n} (x_{i}))$ is a converging sequence with elements from a finite set $S (B)$ , implies $h (x_{i}) \in S (B)$ , $\forall i$ . Thus, $h (S (A)) \subseteq S (B)$ . But since $S (A)$ and $S (B)$ have the same number of elements, $h (S (A)) = S (B)$ .
Definition 2.11
A fuzzy topological space $(Υ, T)$ is said to be fuzzy strongly locally homogeneous abbreviated as $F S L H$ if $U$ is any open set and $x_{a} \in U$ is a fuzzy point, then there exists $V \in T$ such that $x_{a} \in V \subseteq U$ and for any $y_{b} \in V$ , $\exists$ a fuzzy homeomorphism $h$ on $Υ$ mapping $x$ to $y$ and leaving all other points outside $S (V)$ fixed.
Theorem 2.12
If $(Υ, T)$ is fuzzy homogeneous and $T$ contains a fuzzy crisp point, then $Υ$ is $F S L H$ .
Proof.
Let $x_{I} \in T$ be the fuzzy crisp point of $x$ and $y \in Υ$ . Then $\exists$ $h \in H (Υ, T)$ such that $h (x) = y ⟹ y_{I} = h (x_{I}) \in T$ . Thus, all fuzzy crisp points are open. Now, let $U \in T$ be a non-zero open set and $x_{a} \in U$ . Take $V = U \cap x_{I}$ . Then $V$ is an open fuzzy point such that $V (x) = U (x) > a$ and hence $x_{a} \in V$ and for any $y \neq x, V (y) = 0$ . That is, $V$ is the fuzzy point $x_{U (x)}$ and thus $S (V) = {x}$ . Therefore, $Υ$ is $F S L H$ vacuously.
Remark 2.13
Thus a fuzzy homogeneous space $(Υ, T)$ containing an open crisp point is both $F S L H$ and $F S D$ .
Theorem 2.14
$F S L H$ is a fuzzy topological property.
Proof.
Let $(Υ, T_{1})$ and $(Υ^{'}, T_{2})$ be fuzzy topological spaces such that $(Υ, T_{1})$ is $F S L H$ and $f : (Υ, T_{1}) \to (Υ^{'}, T_{2})$ be a fuzzy homeomorphism. Let $V$ be an open set in $Υ^{'}$ and let $y_{b} \in V$ be a fuzzy point. Then $x_{b} = f^{- 1} (y_{b})$ is a fuzzy point in the open set $f^{- 1} (V)$ of $Υ$ and so, $\exists$ an open set $U$ of $Υ$ such that $x_{b} \in U \subseteq f^{- 1} (V)$ and if $p_{t} \in U$ is any distinct fuzzy point, $\exists$ a homeomorphism $h_{p} : (Υ, T_{1}) \to (Υ, T_{1})$ such that $h_{p} (p) = x$ and all other points outside $S (U)$ are fixed by $h_{p}$ . Now $y_{b} = f (x_{b}) \in f (U) \subseteq V$ . Any fuzzy point in $f (U)$ can be written as $f (q_{r})$ with $q_{r} \in U$ , hence
$\begin{aligned} h_{q} (q) = x ⟹ (f \circ h_{q} \circ f^{- 1}) (f (q)) = y \end{aligned}$
Then, $f \circ h_{q} \circ f^{- 1}$ is our required homeomorphism. For,
$\begin{aligned} f (z) \notin S (f (U)) ⟹ & f (z) \notin f (S (U)) \\ ⟹ & z \notin S (U) \\ ⟹ & h_{q} (z) = z \end{aligned}$
Therefore, $(f \circ h_{q} \circ f^{- 1}) (f (z)) = f (z)$ .
3. Almost Fuzzy Countable Dense Homogeneity (AFCDH)

In the ordinary sense, a separable space $Υ$ is said to be countable dense homogeneous ( $C D H$ ) if given any two countable dense subsets $D$ and $E$ of $Υ$ , $\exists$ $h \in H (Υ)$ such that $h (D) = E$ . Later, Al-Ghour and Fora (2016) extended the notion of $C D H$ to fuzzy topological spaces and introduced the concept of fuzzy countable dense homogeneity ( $F C D H$ ). A fuzzy topological space $(Υ, T)$ is $F C D H$ iff it is separable and for any countable dense $(I)$ (see Definition 3.1) collections $D_{1}, D_{2}$ of fuzzy points, $\exists$ $h \in H (Υ)$ such that $h (S (D_{1})) = S (D_{2})$ . Since there are spaces that are not $F C D H$ but are more likely to be, this paper has further extended $F C D H$ to include such spaces that are known as almost fuzzy $C D H$ spaces. For that, let us start with some definitions.

Definition 3.1 Al-Ghour & Fora, 2016

A collection $D$ of fuzzy points in a fuzzy topological space $(Υ, T)$ is said to be dense $(I)$ if for every non-zero $O \in T$ there exists $p \in D$ such that $p \in O$ .

Definition 3.2 Al-Ghour & Fora, 2016

A collection $D$ of fuzzy points in a fuzzy topological space $(Υ, T)$ is said to be dense $(I I)$ if $C l (⋃_{p \in D} p) = 1$ .

It is easy to show that dense $(I)$ and dense $(I I)$ are fuzzy topological properties. The following facts can be observed from Al-Ghour and Fora (2016): The above two definitions are independent of each other. Depending on the presence of dense $(I)$ (or dense $(I I)$ resp.) collection of fuzzy points, the space is separable $(I)$ (or separable $(I I)$ resp.). But finally, it is proven that a fuzzy topological space is separable $(I)$ iff it is separable $(I I)$ . Thus, we simply say that the fuzzy topological space is separable.

Theorem 3.3
Let $(Υ, T)$ be a fuzzy topological space and $D$ be a set of fuzzy points such that $x_{a} \in D$ iff $x_{1 - a} \in D$ . Then $D$ is dense $(I)$ iff $D$ is dense $(I I)$ .
Proof.
Suppose $D$ is dense(I), then $\forall$ open set $O$ , $\exists$ $x_{a} \in D$ such that $x_{a} \in O$ $⟹ a < O (x) ⟹ 1 - a > 1 - O (x)$ . That is, $\forall$ closed set $C \neq 1$ , $\exists$ $x_{b} \in D$ such that $b > C (x)$ or otherwise, $\forall$ closed set $C \neq 1$ , $\exists$ $x \in Υ$ such that $(⋃_{p \in D}^{p}) (x) > C (x)$ . This implies $C l$ ( $⋃_{p \in D} p) = 1$ .

Conversely, let $C l$ ( $⋃_{p \in D} p) = 1$ .
$\begin{aligned} ⟹ & \forall closed set C \neq 1, (⋃_{p \in D} p) ⊈ C \\ ⟹ & \forall closed set C \neq 1, \exists an x \in Υ s.t. (⋃_{p \in D} p) (x) > C (x) \\ ⟹ & \forall closed set C \neq 1, \exists some P = x_{α} \in D s.t. α > C (x) \end{aligned}$
Let $O$ be an open set, then $1 - O$ is closed. This implies $\exists$ $P = y_{β} \in D$ such that $β > 1 - O (y) ⟹ O (y) > 1 - β$ . Thus $y_{1 - β} \in O$ , but $y_{1 - β} \in D$ by the hypothesis.
Definition 3.4
For a non-empty $A \subseteq Υ$ , $Q (A)$ will denote the set
$Q (A) = {fuzzy points x_{r} : x \in A and r \in Q \cap (0, 1)}$

Theorem 3.5 (Al-Ghour & Fora, 2016)

For a fuzzy topological space $(Υ, T)$ and a collection $D$ of fuzzy points of $Υ$ , the following are true:

(i)
If $D$ is dense $(I)$ , then $Q (S (D))$ is dense $(I I)$ .
(ii)
If $D$ is dense $(I I)$ , then $Q (S (D))$ is dense $(I)$ .

Corollary 3.6
Let $D$ be a dense $(I)$ (or dense $(I I)$ collection of fuzzy points. Then $Q (S (D))$ is dense $(I)$ iff $Q (S (D))$ is dense $(I I)$ .
Proof.
Suppose $D$ is dense $(I)$ . Then,
$\begin{aligned} Q (S (D)) is dense (I I) \\ ⟹ Q (S (Q (S (D)))) is dense (I) \\ ⟹ Q (S (D)) is dense (I) \\ ⟹ Q (S (Q (S (D)))) is dense (I I) \\ ⟹ Q (S (D)) is dense (I I) \end{aligned}$

Same is true when $D$ is dense $(I I)$ .
Definition 3.7 Tallafha & Al-Rawashdeh, 2010

In ordinary topology, a separable space $(Υ, T)$ is called an almost countable dense homogeneous space abbreviated as $A C D H$ if for any two countable dense subsets $D_{1}, D_{2}$ , there are two finite subsets $F_{1}, F_{2}$ of $Υ$ , such that $D_{1} \cap F_{1} = D_{2} \cap F_{2} = \emptyset$ and $h \in H (Υ)$ , the space of homeomorphisms on $Υ$ , such that $h (D_{1} \cup F_{1}) = D_{2} \cup F_{2}$ and $h (F_{1}) \cap F_{2} = \emptyset$ .

This notion of almost countable dense homogeneity has been extended to fuzzy topological spaces, giving rise to the concept of almost fuzzy countable dense homogeneity, which forms a proper extension. Studying such spaces is important, as there exist spaces that are not fuzzy countable dense homogeneous ( $F C D H$ ) due to finitely many points, but are AFCDH, as illustrated in Example 3.25.

Definition 3.8
Let $(Υ, T)$ be a fuzzy topological space. Then $Υ$ is said to be almost fuzzy countable dense homogeneous $I$ denoted as $A F C D H (I)$ (or almost fuzzy countable dense homogeneous $I I$ denoted as $A F C D H (I I)$ resp.) if for any two countably dense $(I)$ (or dense $(I I)$ resp.) collection of fuzzy points $D_{1}, D_{2}$ of $Υ$ , there exist two finite subsets of fuzzy points $F_{1}, F_{2}$ of $Υ$ such that (i)
$S (D_{1}) \cap S (F_{1}) = S (D_{2}) \cap S (F_{2}) = \emptyset$ and,
(ii)
$\exists$ $h \in H (Υ)$ satisfying $h (S (D_{1} \cup F_{1})) = S (D_{2} \cup F_{2})$ and $h (S (F_{1})) \cap S (F_{2}) = \emptyset$ .

Theorem 3.9
$A F C D H (I)$ is a fuzzy topological property.
Proof.
Let $(Υ, T_{1})$ and $(Υ^{'}, T_{2})$ be two fuzzy topological spaces and $f : (Υ, T_{1}) \to (Υ^{'}, T_{2})$ be a fuzzy homeomorphism. Suppose $(Υ, T_{1})$ is $A F C D H (I)$ and $D_{1}, D_{2}$ are two countable dense $(I)$ collection of fuzzy points of $Υ^{'}$ . Then so are $f^{- 1} (D_{1}), f^{- 1} (D_{2})$ in $Υ$ . Thus $\exists$ finite sets $F_{1}, F_{2}$ of fuzzy points of $Υ$ such that $S (f^{- 1} (D_{1})) \cap S (F_{1}) = S (f^{- 1} (D_{2})) \cap S (F_{2}) = \emptyset$
$\begin{aligned} ⟹ & f^{- 1} (S (D_{1})) \cap f^{- 1} (S (f (F_{1}))) = f^{- 1} (S (D_{2})) \cap f^{- 1} (S (f (F_{2}))) = \emptyset \\ ⟹ & f^{- 1} (S (D_{1}) \cap S (f (F_{1}))) = f^{- 1} (S (D_{2}) \cap S (f (F_{2}))) = \emptyset \\ ⟹ & S (D_{1}) \cap S (f (F_{1})) = S (D_{2}) \cap S (f (F_{2})) = \emptyset \end{aligned}$
and $\exists$ $h \in H (Υ)$ such that $h (S (F_{1})) \cap S (F_{2}) = \emptyset$ and $h (S (f^{- 1} (D_{1}) \cup F_{1})) = S (f^{- 1} (D_{2}) \cup F_{2})$ .
$\begin{aligned} ⟹ & h (f^{- 1} (S (D_{1}) \cup f (S (F_{1})))) = f^{- 1} (S (D_{2}) \cup f (S (F_{2}))) \\ ⟹ & (f \circ h \circ f^{- 1}) (S (D_{1} \cup f (F_{1}))) = S (D_{2} \cup f (F_{2})) \end{aligned}$

Theorem 3.10
A fuzzy topological space $(Υ, T)$ is $A F C D H (I)$ iff it is $A F C D H (I I)$ .
Proof.
Let $(Υ, T)$ be $A F C D H (I)$ and $D_{1}, D_{2}$ be two countably dense $(I I)$ collection of fuzzy points of $Υ$ . Then by Theorem 3.5, $Q (S (D_{1}))$ and $Q (S (D_{2}))$ are dense $(I)$ collection of fuzzy points and thus there exists two finite subsets of fuzzy points $F_{1}, F_{2}$ such that
$\begin{aligned} \begin{aligned} S (Q (S (D_{1}))) \cap S (F_{1}) = S (Q (S (D_{2}))) \cap S (F_{2}) = \emptyset \\ ⟹ & S (D_{1}) \cap S (F_{1}) = S (D_{2}) \cap S (F_{2}) = \emptyset, since S (Q (S (D_{i}))) = S (D_{i}) \end{aligned} \end{aligned}$
and there exists a homeomorphism $h \in H (Υ)$ satisfying
$\begin{aligned} h (S (Q (S (D_{1})) \cup F_{1})) = S (Q (S (D_{2})) \cup F_{2}) and \\ h (S (F_{1})) \cap S (F_{2}) = \emptyset \\ ⟹ & h (S (D_{1} \cup F_{1})) = S (D_{2} \cup F_{2}) \end{aligned}$
Thus, $Υ$ is $A F C D H (I I)$ . The converse is similar.
Theorem 3.11
In fuzzy topological spaces, $A F C D H (I I)$ is a fuzzy topological property.
Proof.
Let $(Υ, T_{1})$ and $(Υ^{'}, T_{2})$ be fuzzy topological spaces such that $(Υ, T_{1})$ is $A F C D H (I I)$ and let $h : (Υ, T_{1}) \to (Υ^{'}, T_{2})$ be a fuzzy homeomorphism. By Theorem 3.10, $(Υ, T_{1})$ is $A F C D H (I)$ and by Theorem 3.9, $(Υ^{'}, T_{2})$ is $A F C D H (I)$ . Again by Theorem 3.10, $(Υ^{'}, T_{2})$ is $A F C D H (I I)$ .
Remark 3.12
From here onward, we do not distinguish between $A F C D H (I)$ and $A F C D H (I I)$ , instead we simply say the space is $A F C D H$ . In general, $A F C D H$ spaces need not be $T_{1}$ . For example, consider $Υ = {1, 2}$ and $T = {0, 1, p_{a}, q_{b}, λ}$ where $p_{a}$ and $q_{b}$ are two fuzzy points with $p = 1, q = 2, a = 0.1, b = 0.2$ and $λ = {(1, 0.1), (2, 0.2)}$ (i.e. $λ$ maps 1 to 0.1 and 2 to 0.2). Considering $p_{c}, q_{d}$ with $c = 0.2, d = 0.3$ shows that $Υ$ is not $T_{1}$ . But clearly it is $A F C D H$ , since any countable dense collection $D$ of fuzzy points contains $p_{< a}$ and $q_{< b}$ (where $< a, < b$ denotes any positive value less than $a$ and $b$ respectively) and thus the identity map will be the required homeomorphism.
Proposition 3.13
If $Υ$ is a countable $A F C D H$ space and $D$ is any dense collection of fuzzy points of $Υ$ , then $Υ ∖ S (D)$ is finite.
Proof.
Suppose that $Υ ∖ S (D)$ is countably infinite. Since $Q (Υ)$ is a countable dense collection of fuzzy points, $\exists$ finite sets $F_{1}, F_{2}$ and a homeomorphism $h : (Υ, T) \to (Υ, T)$ satisfying $A F C D H$ conditions, particularly $h (S (D \cup F_{1})) = S (Q (Υ) \cup F_{2}) = Υ$ , a contradiction since $S (D \cup F_{1}) ⊊ Υ$ .
Definition 3.14
Let $(Υ, T)$ be a fuzzy topological space and $A$ be the collection of all open fuzzy points of $Υ$ . Then, $Υ$ is said to be almost fuzzy semi discrete, denoted by $A F S D$ if $Υ ∖ S (A)$ is finite.
Example 3.15
Clearly, every finite fuzzy topological space is $A F S D$ .

By definition, every semi discrete space is almost semi discrete and for an $A F C D H$ space $Υ$ to be $A F S D$ , $Υ$ can be at the most countably infinite only.
Theorem 3.16
Let $(Υ, T)$ be a countable fuzzy separable space. Then $Υ$ is $A F S D$ $⟺$ $Υ$ is $A F C D H$ .
Proof.
Let $Υ$ be fuzzy separable and $A$ be the collection of all open fuzzy points of $Υ$ . Suppose that $Υ$ is $A F S D$ , then $Υ ∖ S (A) = F$ , a finite set. Let $D_{1}$ and $D_{2}$ be two countable dense collections of fuzzy points of $Υ$ . Then clearly $Υ ∖ F \subseteq S (D_{1})$ and $Υ ∖ F \subseteq S (D_{2})$ . Choose finite sets of fuzzy points $F_{1}$ and $F_{2}$ such that $S (F_{1}) = S (D_{2}) ∖ S (D_{1})$ and $S (F_{2}) = S (D_{1}) ∖ S (D_{2})$ . Then, $S (D_{1}) \cap S (F_{1}) = S (D_{2}) \cap S (F_{2}) = \emptyset$ and the identity homeomorphism $I$ satisfies $I (S (D_{1} \cup F_{1})) = S (D_{2} \cup F_{2})$ and $I (S (F_{1})) \cap S (F_{2}) = \emptyset$ .

Conversely, suppose that $Υ ∖ S (A)$ is infinite. Since $Υ$ is $A F C D H$ and $A$ is a countable dense set of fuzzy points, there exists a finite set $F$ of fuzzy points and $h \in H (Υ)$ such that $h (S (Q (Υ)) = S (A \cup F) ⟹ Υ = S (A \cup F)$ , which is a contradiction.
Definition 3.17
In $A F C D H$ spaces, if cardinality of $F_{1}$ and $F_{2}$ are the same, such spaces are said to be strong almost fuzzy countable dense homogeneous, abbreviated as $S A F C D H$ .
Theorem 3.18
A fuzzy topological space $(Υ, T)$ is $S A F C D H$ if and only if for every two countable dense sets $D_{1}, D_{2}$ of fuzzy points, there exist two finite subsets $F_{1}, F_{2}$ of fuzzy points having the same cardinality such that $S (F_{1}) \subseteq S (D_{1}), S (F_{2}) \subseteq S (D_{2})$ and there exists $h \in H (Υ)$ such that $h (S (D_{1}) ∖ S (F_{1})) = S (D_{2}) ∖ S (F_{2})$ .
Proof.
Let $D_{1}, D_{2}$ be two countable dense sets of fuzzy points. Suppose that there exist two finite subsets $G_{1}, G_{2}$ of fuzzy points of the same cardinality such that $S (G_{1}) \subseteq S (D_{1}), S (G_{2}) \subseteq S (D_{2})$ and $\exists$ $h \in H (Υ)$ such that $h (S (D_{1}) ∖ S (G_{1})) = S (D_{2}) ∖ S (G_{2})$ . Let $S (F_{1}) = h^{- 1} (S (G_{2})) ∖ S (D_{1})$ and $S (F_{2}) = h (S (G_{1})) ∖ S (D_{2})$ . It is clear that $S (F_{1}) \cap S (D_{1}) = S (F_{2}) \cap S (D_{2}) = \emptyset$ . Also,
$\begin{aligned} ∣ S (F_{1}) ∣ & = ∣ h^{- 1} (S (G_{2})) ∖ S (D_{1}) ∣ \\ = ∣ h^{- 1} (S (G_{2})) ∖ (S (D_{1}) \cap h^{- 1} (S (G_{2}))) ∣ \\ = ∣ h^{- 1} (S (G_{2})) ∣ - ∣ S (D_{1}) \cap h^{- 1} (S (G_{2})) ∣ \\ = ∣ S (G_{1}) ∣ - ∣ h (S (D_{1})) \cap S (G_{2}) ∣ \\ = ∣ S (G_{1}) ∣ - ∣ h (S (D_{1}) ∖ S (G_{1}) \cup S (G_{1})) \cap S (G_{2}) ∣ \\ = ∣ S (G_{1}) ∣ - ∣ h (S (G_{1})) \cap S (G_{2}) ∣ \\ = ∣ S (G_{1}) ∣ - ∣ (S (G_{2}) \cap h (S (G_{1}))) \cup ((S (D_{2}) ∖ S (G_{2})) \cap h (S (G_{1}))) ∣ \\ = ∣ h (S (G_{1})) ∣ - ∣ S (D_{2}) \cap h (S (G_{1})) ∣ \\ = ∣ h (S (G_{1})) ∖ S (D_{2}) \cap h (S (G_{1})) ∣ \\ = ∣ S (F_{2}) ∣ \end{aligned}$
And,
$\begin{aligned} h (S (D_{1} \cup F_{1})) & = h (S (D_{1})) \cup h (S (F_{1})) \\ = h (S (D_{1})) \cup S (G_{2}) ∖ h (S (D_{1})) \\ = h (S (D_{1})) \cup S (G_{2}) \\ = h (S (D_{1}) ∖ S (G_{1})) \cup h (S (G_{1})) \cup S (G_{2}) \\ = S (D_{2}) ∖ S (G_{2}) \cup h (S (G_{1})) \\ = S (D_{2}) \cup h (S (G_{1})) \\ = S (D_{2}) \cup h (S (G_{1})) ∖ S (D_{2}) \cup S (D_{2}) \\ = S (D_{2}) \cup S (F_{2}) \end{aligned}$
Finally, $h (S (F_{1})) \cap S (F_{2}) = \emptyset$ . For if, $x \in h (S (F_{1})) \cap S (F_{2}) ⟹ x \in S (G_{2}) ∖ h (S (D_{1})) and x \in h (S (G_{1})) ∖ S (D_{2}) ⟹ x \in S (G_{2}) and x \notin S (D_{2})$ , not possible. Thus $Υ$ is $S A F C D H$ .

Conversely, let $D_{1}, D_{2}$ be two countable dense sets of fuzzy points. Then there exist two finite subsets $G_{1}, G_{2}$ of fuzzy points of the same cardinality such that $S (G_{1}) \cap S (D_{1}) = S (G_{2}) \cap S (D_{2}) = \emptyset$ and $\exists$ $h \in H (Υ)$ such that $h (S (D_{1} \cup G_{1})) = S (D_{2} \cup G_{2})$ and $h (S (G_{1})) \cap S (G_{2}) = \emptyset$ . Take $S (F_{1}) = h^{- 1} (S (G_{2}))$ and $S (F_{2}) = h (S (G_{1}))$ . Then $S (F_{1}) \subseteq S (D_{1}), S (F_{2}) \subseteq S (D_{2}), ∣ S (F_{1}) ∣=∣ S (F_{2}) ∣$ and
$\begin{aligned} h (S (D_{1}) ∖ S (F_{1})) & = h (S (D_{1}) ∖ h^{- 1} (S (G_{2}))) \\ = h (S (D_{1})) ∖ S (G_{2}) \\ = h (S (D_{1} \cup G_{1}) ∖ S (G_{1})) ∖ S (G_{2}) \\ = h (S (D_{1} \cup G_{1})) ∖ (h (S (G_{1})) \cup S (G_{2})) \\ = S (D_{2}) \cup S (G_{2}) ∖ (h (S (G_{1})) \cup S (G_{2})) \\ = S (D_{2}) ∖ h (S (G_{1})) \\ = S (D_{2}) ∖ S (F_{2}) \end{aligned}$

Remark 3.19
In contrast to $A F C D H$ spaces, we could think of the deletion of finitely many points from the spaces $Υ$ and $Υ^{'}$ , so that they can be countable dense homogeneous and this provides the rationale for the following definition.
Definition 3.20
A separable fuzzy topological space $(Υ, T)$ is called an almost fuzzy $C D H$ of type , denoted as $A F C D H^{}$ if there exist a finite subset $F$ of fuzzy points such that for any countable dense subsets $A$ and $B$ of fuzzy points, $\exists$ $h \in H (Υ)$ such that
$h (S (A) ∖ S (F)) = S (B) ∖ S (F)$

This $F$ is referred as a related fuzzy finite set.
Proposition 3.21
Let $(Υ, T)$ be a countable $A F C D H^{}$ space with a related fuzzy finite set $F$ , then $\forall$ $x \notin S (F)$ , $x_{a}$ is open for some $a \in (0, 1]$ .
Proof.
Let $x \notin S (F)$ and suppose that $x_{a}$ is not open $\forall$ $a \in (0, 1]$ . Then both $Q (Υ ∖ {x})$ and $Q (Υ)$ are countable dense subsets of fuzzy points and so $\exists$ $h \in H (Υ)$ such that
$\begin{aligned} h (S (Q (Υ)) ∖ S (F)) = S (Q (Υ ∖ {x})) ∖ S (F) \\ ⟹ & h (Υ ∖ S (F)) = (Υ ∖ {x}) ∖ S (F) \end{aligned}$
which is a contradiction.
Theorem 3.22
Let $(Υ, T)$ be a countable fuzzy topological space. Then $Υ$ is $A F C D H^{}$ iff $Υ$ is $A F S D$ .
Proof.
The first part easily follows from the above proposition.

Conversely, let $A$ be the set of all open fuzzy points, then $Υ ∖ S (A) = D$ is finite. Let $D_{1}$ and $D_{2}$ be any two countable dense subsets of fuzzy points. Clearly, $Υ ∖ D \subseteq S (D_{1})$ and $Υ ∖ D \subseteq S (D_{2})$ . Choose $F$ such that $S (F) = D$ , then $S (D_{i}) ∖ S (F) = Υ ∖ D$ for $i = 1, 2$ and hence the identity map will be the required homeomorphism for $(Υ, T)$ to be $A F C D H^{}$ .
Corollary 3.23
A countable fuzzy topological space $(Υ, T)$ is $A F C D H$ iff $(Υ, T)$ is $A F C D H^{}$ .
Proof.
By Theorem 3.16, a countable fuzzy space $Υ$ is $A F C D H$ iff $Υ$ is $A F S D$ and by Theorem 3.22, $Υ$ is $A F S D$ $⟺$ it is $A F C D H^{}$ .
Remark 3.24
Therefore, by Theorems 3.16 and 3.22, a countable fuzzy separable space $Υ$ is $A F S D$ $⟺$ $Υ$ is $A F C D H$ or $A F C D H^{}$ .
Example 3.25
The following is an example of a fuzzy space that is not $F C D H$ but is $A F C D H$ . Consider the space $(Υ, T)$ with $Υ = {1, 2, 3}$ and $T = {0, 1, λ}$ where $λ = {(1, 0.1), (2, 0.1), (3, 0.1)}$ . Let $D_{1} = {1_{0.01}}$ and $D_{2} = {1_{0.01}, 2_{0.02}}$ . Then both are countable dense sets of fuzzy points with $S (D_{1}) = {1}$ and $S (D_{2}) = {1, 2}$ . Clearly $S (D_{1})$ and $S (D_{2})$ are not homeomorphic, and thus the space is not $F C D H$ . If we take $S (F) = {2}$ , then $S (D_{1}) ∖ S (F)$ and $S (D_{2}) ∖ S (F)$ are homeomorphic by the trivial homeomorphism. Similarly, for any $D_{1}$ and $D_{2}$ , choose $F$ appropriately so that cardinality of $S (D_{1}) ∖ S (F)$ and $S (D_{2}) ∖ S (F)$ are the same, then, any bijection would be a homeomorhism on $Υ$ mapping $S (D_{1}) ∖ S (F)$ to $S (D_{2}) ∖ S (F)$ . Hence $Υ$ is $A F C D H$ by the above Corollary.
4. Fuzzy Transposition Homogeneity

A transposition on a non-empty set $Υ$ refers to a permutation on $Υ$ that swaps the positions of two elements and leaves all other elements unaffected. An ordinary topological space $(Υ, T)$ is called transposition-homogeneous abbreviated as $T H$ if every transposition on $Υ$ is a homeomorphism (Tallafha & Al-Rawashdeh, 2010). A corresponding formulation can be introduced in fuzzy topological spaces, yielding a proper extension of the classical concept.

Definition 4.1
A fuzzy topological space $(Υ, T)$ is said to be fuzzy transposition homogeneous denoted by $F T H$ , if every transposition on $Υ$ is a fuzzy homeomorphism.
Definition 4.2 (Jose & Mathew, 2011)

An fuzzy topological space $(Υ, T)$ is said to be strongly homogeneous if for every pair of elements $x, y \in Υ$ there exists a homeomorphism $h$ such that $h (x) = y$ and $h (y) = x$ .

Remark 4.3
It is evident that $F T H$ is a stronger notion than strong homogeneity. Hence, every fuzzy transposition homogeneous space is necessarily strongly homogeneous.
Theorem 4.4
In an $F T H$ space $(Υ, T)$ , any finite composition of transpositions of an open set is open.
Proof.
If $g \in T$ , then any $f \in [0, 1]^{Υ}$ that agrees with $g$ at all except at two points is an open set, since $Υ$ is $F T H$ . Let $F \subseteq Υ$ be any finite set and $σ$ be any permutation that fixes $Υ ∖ F$ . Clearly $σ$ is a composition of finitely many transpositions which are homeomorphisms, and thus $σ$ is a homeomorphism. Therefore, for any open set $g$ , an $f \in [0, 1]^{Υ}$ that agrees with $g$ at all but a finite number of points is also open.
Example 4.5
Let $Υ = {a, b, c}$ and $f \in [0, 1]^{Υ}$ be defined by
$\begin{aligned} f (x) = {\begin{cases} \frac{1}{2} & x = a \\ \frac{1}{3} & x = b \\ \frac{1}{4} & x = c \end{cases} \end{aligned}$
In short, let $f = {\frac{1}{2}, \frac{1}{3}, \frac{1}{4}}$ . By Theorem 4.4, a topology $T$ for which $Υ$ is $F T H$ and containing $f$ , shall contain all permutations of $f$ , their unions, and intersections. Thus, $T$ is given by
$\begin{aligned} T & = {0, 1, {\frac{1}{2}, \frac{1}{3}, \frac{1}{4}}, {\frac{1}{3}, \frac{1}{2}, \frac{1}{4}}, {\frac{1}{4}, \frac{1}{3}, \frac{1}{2}}, {\frac{1}{2}, \frac{1}{4}, \frac{1}{3}}, \\ {\frac{1}{3}, \frac{1}{4}, \frac{1}{2}}, {\frac{1}{4}, \frac{1}{2}, \frac{1}{3}}, {\frac{1}{4}, \frac{1}{3}, \frac{1}{3}}, {\frac{1}{3}, \frac{1}{3}, \frac{1}{4}}, {\frac{1}{3}, \frac{1}{4}, \frac{1}{3}}, \\ {\frac{1}{3}, \frac{1}{4}, \frac{1}{4}}, {\frac{1}{4}, \frac{1}{3}, \frac{1}{4}}, {\frac{1}{4}, \frac{1}{4}, \frac{1}{3}}, {\frac{1}{2}, \frac{1}{4}, \frac{1}{4}}, {\frac{1}{4}, \frac{1}{2}, \frac{1}{4}}, \\ {\frac{1}{4}, \frac{1}{4}, \frac{1}{2}}, {\frac{1}{2}, \frac{1}{2}, \frac{1}{4}}, {\frac{1}{2}, \frac{1}{4}, \frac{1}{2}}, {\frac{1}{4}, \frac{1}{2}, \frac{1}{2}}, {\frac{1}{2}, \frac{1}{3}, \frac{1}{2}}, \\ {\frac{1}{3}, \frac{1}{2}, \frac{1}{2}}, {\frac{1}{2}, \frac{1}{2}, \frac{1}{3}}, {\frac{1}{2}, \frac{1}{2}, \frac{1}{2}}, {\frac{1}{3}, \frac{1}{3}, \frac{1}{3}}, {\frac{1}{4}, \frac{1}{4}, \frac{1}{4}}} \end{aligned}$

Lemma 4.6
An $F T H$ space is $F S D$ iff either a fuzzy point or a fuzzy crisp point is open.
Proof.
By the above theorem, for an $F T H$ space $(Υ, T)$ , if $x_{a} \in T$ is any fuzzy point, then $y_{a} \in T$ for every $y \in Υ$ making the space semi-discrete. The same is the case if $T$ contains any crisp point.
Theorem 4.7
A fuzzy semi-discrete $F T H$ space is $F C D H$ iff it is countable.
Proof.
The first part is immediate.

Conversely, suppose that $Υ$ is countable. Then, by $F S D$ property, for any countable dense $(I)$ collection $D$ of fuzzy points of $Υ$ , $S (D) = Υ$ making the space $F C D H$ using the identity map.
Theorem 4.8
$F T H$ is a fuzzy topological property.
Proof.
Let $(Υ, T_{1}), (Υ^{'}, T_{2})$ be two fuzzy topological spaces, and $f : (Υ, T_{1}) \to (Υ^{'}, T_{2})$ be a fuzzy homeomorphism. Suppose that $(Υ, T_{1})$ is $F T H$ . Let $y_{1}, y_{2} \in Υ^{'}$ and $h : (Υ^{'}, T_{2}) \to (Υ^{'}, T_{2})$ be defined by $h (y_{1}) = y_{2}$ , $h (y_{2}) = y_{1}$ and $h (y) = y$ , $\forall y \neq y_{1}, y_{2} \in Υ^{'}$ . It is enough to show that $h$ is a fuzzy homeomorphism. Clearly, it is a bijection. Let $g \in T_{2}$ . Now, $h (g) \in T_{2} ⟺ f^{- 1} (h (g)) \in T_{1}$ . By definition,
$\begin{aligned} h (g) (y) & = {\begin{cases} g (y_{2}) & y = y_{1} \\ g (y_{1}) & y = y_{2} \\ g (y) & y \neq y_{1}, y_{2} \end{cases} \end{aligned}$

$\begin{aligned} ⟹ f^{- 1} (h (g)) (y) & = {\begin{cases} f^{- 1} (g) (y_{2}) & y = y_{1} \\ f^{- 1} (g) (y_{1}) & y = y_{2} \\ f^{- 1} (g) (y) & y \neq y_{1}, y_{2} \end{cases} \end{aligned}$
i.e., $f^{- 1} (h (g))$ is a transposition of $f^{- 1} (g)$ . But, $g \in T_{2}$ $⟹$ $f^{- 1} (g) \in T_{1}$ and hence $f^{- 1} (h (g)) \in T_{1}$ , since $Υ$ is $F T H$ .
Theorem 4.9
An $F T H$ space $(Υ, T)$ is $T_{1}$ iff either a fuzzy crisp point is open or the set $A_{x y} = {a \in (0, 1) ∣ f_{a} \in T; f_{a} (x) = 0, f_{a} (y) > a}$ for some $x, y \in Υ$ , is dense in $(0, 1)$ .
Proof.
Suppose $(Υ, T)$ is $T_{1}$ . Then for any two fuzzy points $y_{a}, x_{b}$ , there exists open sets $U, V$ such that $a < U (y), b < V (x)$ and $U (x) = V (y) = 0$ . Suppose that no fuzzy crisp points are open. Then $f_{a} = U$ . Since such $f_{a}$ ’s exists for all $a \in (0, 1)$ but can coincide at infinitely many ’ $a$ ’ values, it follows that $A_{x y}$ is dense in $(0, 1)$ .

Conversely, suppose that $T$ contains a fuzzy crisp point. Since $Υ$ is $F T H$ , all fuzzy crisp points are open and hence $Υ$ is $T_{1}$ . Suppose that for some ordered pair $x, y \in Υ$ , the set $A_{x y}$ is dense in $(0, 1)$ , then, $F T H$ property of $Υ$ implies $A_{x^{'} y^{'}}$ is dense in $(0, 1)$ for any ordered pair $x^{'}, y^{'} \in Υ$ . Thus for any fuzzy points $y_{a}, x_{b}$ , there exists $f_{a^{'}} \in A_{x y}, f_{b^{'}} \in A_{y x}$ such that
$\begin{aligned} f_{a^{'}} (x) = 0, f_{a^{'}} (y) > a^{'} \geq a and \\ f_{b^{'}} (y) = 0, f_{b^{'}} (x) > b^{'} \geq b \end{aligned}$
Thus $x_{b} \in f_{b^{'}}$ and $y_{a} \in f_{a^{'}}$ .
Theorem 4.10
A fuzzy topological space $(Υ, T)$ is $F T H$ iff for any two finite subsets $F_{1}, F_{2}$ of fuzzy points having the same cardinality and for every $h \in H (Υ)$ , there exists $h^{'} \in H (Υ)$ such that (i)
$h^{'} (S (F_{1})) = S (F_{2})$
(ii)
$h^{'} (p) = h (p)$ , $\forall$ $p \notin S (F_{1}) \cup S (h^{- 1} (F_{2}))$
(iii)
$h^{'} (h^{- 1} (S (F_{2}))) = h (S (F_{1}))$

Proof.
Let $Υ$ be $F T H$ and $h$ be any homeomorphism, then for any transposition $g$ , $g \circ h$ is a homeomorphism. This is true even if $g$ is a finite composition of transpositions. i.e., if $F_{1}$ and $F_{2}$ are finite sets of fuzzy points of the same size, then for any bijective map $g$ for which $g (h (p)) = h (p)$ $\forall p \notin$ finite set $A = S (F_{1}) \cup S (h^{- 1} (F_{2}))$ is a homeomorphism. The image of $A$ under $h$ is $h (S (F_{1})) \cup S (F_{2})$ .
$\begin{aligned} i . e ., h (S (F_{1}) \cup S (h^{- 1} (F_{2}))) = h (S (F_{1})) \cup S (F_{2}) \\ ⟹ & g (h (S (F_{1}) \cup S (h^{- 1} (F_{2})))) = h (S (F_{1})) \cup S (F_{2}), \\ since g (h (A)) = h (A) \end{aligned}$
In particular, $g \circ h$ can map $S (F_{1})$ to $S (F_{2})$ and $S (h^{- 1} (F_{2}))$ or $h^{- 1} (S (F_{2}))$ to $h (S (F_{1}))$ . Thus the first part of the proof is done by renaming $g \circ h$ by $h^{'}$ .

Converse is obvious if $S (F_{1}) = {x}, S (F_{2}) = {y}$ and $h = I$ , the identity map.
Theorem 4.11
If $(Υ, T)$ is an $S A F C D H$ , $F T H$ space, then $(Υ, T)$ is a $F C D H$ space.
Proof.
Let $D_{1}, D_{2}$ be two countable dense subsets of fuzzy points of $Υ$ . Then by Theorem 3.18, there exist two finite sets $F_{1}, F_{2}$ of fuzzy points having the same cardinality such that $S (F_{1}) \subseteq S (D_{1}), S (F_{2}) \subseteq S (D_{2})$ and there exists $h \in H (Υ)$ such that $h (S (D_{1}) ∖ S (F_{1})) = S (D_{2}) ∖ S (F_{2})$ . Also, since $Υ$ is $F T H$ , there exists $h^{'} \in H (Υ)$ such that $h^{'} (S (F_{1})) = S (F_{2})$ , $h^{'} (p) = h (p)$ , for every $p \notin S (F_{1} \cup h^{- 1} (F_{2}))$ and $h^{'} (h^{- 1} (S (F_{2}))) = h (S (F_{1}))$ . We shall prove that $h^{'} (S (D_{1})) = S (D_{2})$ . It is enough to show that $S (D_{1}) \cap h^{- 1} (S (F_{2})) \subseteq S (F_{1})$ . If possible, suppose that $x \in (S (D_{1}) \cap h^{- 1} (S (F_{2}))) ∖ S (F_{1}) ⟹ x \in S (D_{1}) ∖ S (F_{1}) and x \in h^{- 1} (S (F_{2})) ∖ S (F_{1})$ . Then former part implies $h (x) \in S (D_{2}) ∖ S (F_{2})$ , a contradiction to the latter part. Finally, $h^{'} (S (D_{1})) = h^{'} (S (D_{1}) ∖ S (F_{1})) \cup h^{'} (S (F_{1})) = h (S (D_{1}) ∖ S (F_{1})) \cup S (F_{2}) = (S (D_{2}) ∖ S (F_{2})) \cup S (F_{2}) = S (D_{2})$ .
Lemma 4.12 (Al-Ghour & Fora, 2016)

For an ordinary topological space $(Υ, T)$ , $A \subseteq Υ$ and a collection $D$ of fuzzy points of $Υ$ , we have

(i)
If $A$ is dense in $(Υ, T)$ , then $Q (A)$ is dense $(I)$ in $(Υ, ω (T))$ .
(ii)
If $D$ is dense $(I)$ in $(Υ, ω (T))$ , then $S (D)$ is dense in $(Υ, T)$ .

The above lemma provides the essential link between the almost $C D H$ property of the ordinary space $(Υ, T)$ and that of its corresponding fuzzy topological space $(Υ, ω (T))$ , which is stated in the following theorem.
Theorem 4.13
Let $(Υ, T)$ be an ordinary topological space. Then $(Υ, T)$ is $A C D H$ iff $(Υ, ω (T))$ is $A F C D H$ .
Proof.
Let $(Υ, T)$ be $A C D H$ and $D_{1}, D_{2}$ be two countable dense(I) collection of fuzzy points of $(Υ, ω (T))$ . Then, $S (D_{1}), S (D_{2})$ are countable dense subsets of $(Υ, T)$ and so there exists finite sets $G_{1}, G_{2}$ such that $S (D_{1}) \cap G_{1} = S (D_{2}) \cap G_{2} = \emptyset$ and $\exists$ $h \in H (Υ)$ such that $h (S (D_{1}) \cup G_{1}) = S (D_{2}) \cup G_{2}$ and $h (G_{1}) \cap G_{2} = \emptyset$ . Let $F_{1}, F_{2}$ be two finite sets of fuzzy points such that $S (F_{1}) = G_{1}$ and $S (F_{2}) = G_{2}$ . Then, we have $h (S (D_{1}) \cup S (F_{1})) = S (D_{2}) \cup S (F_{2})$ , $h (S (F_{1})) \cap S (F_{2}) = \emptyset$ and $S (D_{1}) \cap S (F_{1}) = S (D_{2}) \cap S (F_{2}) = \emptyset$ . Thus, $(Υ, ω (T))$ is $A F C D H$ .

The converse is similar.
5. Ordinary Topologies Generated by Fuzzy Topologies

This section deals with the study of topological spaces that are generated by fuzzy topological spaces. Further, we shall observe that whenever the fuzzy topological space $(Υ, T)$ is $A F S D$ , or $F T H$ or $A F C D H$ , then so is the corresponding ordinary space $(Υ, T_{0})$ (See Definition 5.1). Now, let us start with the following definition:

Definition 5.1 Al-Ghour & Fora, 2016

Let $(Υ, T)$ be a fuzzy topological space and $a \in [0, 1)$ . Then, $T_{a} = {μ^{- 1} (a, 1] : μ \in T}$ is an ordinary topology on $Υ$ called as $a$ -cut topological space of $(Υ, T)$ .

Clearly, $T_{0}$ is the collection of all supports of open sets.

Theorem 5.2
If $(Υ, T)$ is $A F S D$ space, then $(Υ, T_{0})$ is $A S D$ .
Proof.
If $D \subseteq T$ is the collection of all fuzzy points, then $K = Υ ∖ S (D)$ is finite $⟹$ $\forall x \notin K, {x} \in T_{0}$ .
Remark 5.3
It is obvious that the above result need not be true for $(Υ, T_{a})$ with $a > 0$ . But in certain cases $(Υ, T_{a})$ can be semi discrete where as the space $(Υ, T_{0})$ is not $A S D$ . For example, let $Υ = {1, 2, 3, \dots}$ , the set of all natural numbers and $T$ be the smallest topology containing the set
$\begin{aligned} {1, 0, ((1, 0.6), (2, 0.4), (3, 0.4)), ((1, 0.4), (2, 0.6), (3, 0.4))} \\ \cup {((1, 0.4), (2, 0.4), (3, 0.6)), ((1, 0.4), (2, 0.4), (4, 0.6)), \\ ((1, 0.4), (2, 0.4), (5, 0.6)), ((1, 0.4), (2, 0.4), (6, 0.6)), \dots} \end{aligned}$
Clearly $T$ does not contain fuzzy points and, $T_{0}$ contains a singleton iff $T$ contains a fuzzy point. Where as $T_{0.5}$ contains all singletons of $Υ$ , making $(Υ, T_{0.5})$ semi-discrete.
Theorem 5.4
For a fuzzy topological space $(Υ, T)$ , if $f$ is a fuzzy continuous(homeomorphism) map on $(Υ, T)$ , then $f$ is continuous(homeomorphism) on $(Υ, T_{a})$ for all $a \in [0, 1)$ .
Proof.
Let $U \in T_{a}$ . Then $\exists$ $λ \in T$ such that $U = λ^{- 1} (a, 1]$ . It is enough to show that $f^{- 1} (λ^{- 1} (a, 1]) \in T_{a}$ . But $λ \in T ⟹ f^{- 1} (λ) \in T ⟹ (f^{- 1} (λ))^{- 1} (a, 1] \in T_{a}$ and $(f^{- 1} (λ))^{- 1} (a, 1] = f^{- 1} (λ^{- 1} (a, 1])$ . For,
$\begin{aligned} x \in f^{- 1} (λ^{- 1} (a, 1]) \\ ⟺ f (x) \in λ^{- 1} (a, 1] \\ ⟺ λ (f (x)) \in (a, 1] \\ ⟺ λ (y) \in (a, 1] if f (x) = y \\ ⟺ f^{- 1} (λ) (x) \in (a, 1] \\ ⟺ x \in (f^{- 1} (λ))^{- 1} (a, 1] \end{aligned}$
Thus, for any continuous $f : (Υ, T) \to (Υ, T) ⟹$ $f : (Υ, T_{a}) \to (Υ, T_{a})$ is continuous. The homeomorphism part is straightforward.

The converse of the above need not be true in general. For, let $Υ = {a, b, c}$ and $T = {0, 1, λ_{1}, λ_{2}}$ where $λ_{1} = {(a, \frac{1}{3}), (b, \frac{1}{4}), (c, \frac{1}{2})}, λ_{2} = {(a, \frac{1}{4}), (b, \frac{1}{4}), (c, \frac{1}{4})$ . Then $(Υ, T)$ is a fuzzy topological space, and for $a = \frac{2}{5}, T_{a} = {\emptyset, Υ, {c}}$ . Define $h : (Υ, T_{\frac{2}{5}}) \to (Υ, T_{\frac{2}{5}})$ by $h (a) = b, h (b) = a, h (c) = c$ . Clearly $h$ is a homeomorphism on $T_{\frac{2}{5}}$ . But $h^{- 1} (λ_{1}) = {(a, \frac{1}{4}), (b, \frac{1}{3}), (c, \frac{1}{2})} \notin T$ .
Corollary 5.5
If $(Υ, T)$ is $F T H$ , then $(Υ, T_{0})$ is $T H$ .
Proof.
Let $(Υ, T)$ be $F T H$ . Then every transposition on $Υ$ is a fuzzy homeomorphism and by Theorem 5.4 , they are also homeomorphisms on $T_{0}$ . This implies $(Υ, T_{0})$ is $T H$ .
Lemma 5.6 (Al-Ghour & Fora, 2016)

For a fuzzy topological space $(Υ, T)$ , $A \subseteq Υ$ and a collection $D$ of fuzzy points of $Υ$ . Then we have,

(i)
If $A$ is dense in $(Υ, T_{0})$ , then $Q (A)$ is dense in $(Υ, T)$ .
(ii)
If $D$ is dense in $(Υ, T)$ , then $S (D)$ is dense in $(Υ, T_{0})$ .

Theorem 5.7
If $(Υ, T)$ is an $A F C D H$ space, then $(Υ, T_{0})$ is $A C D H$ .
Proof.
Let $(Υ, T)$ be $A F C D H$ then, by Lemma 5.6 $(Υ, T_{0})$ is separable. Let $A$ and $B$ be two countable dense subsets of $(Υ, T_{0})$ then, again by Lemma 5.6 $Q (A), Q (B)$ are countable and dense in $(Υ, T)$ . So $\exists$ finite subsets $F_{1}, F_{2}$ of fuzzy points satisfying
$\begin{aligned} S (Q (A)) \cap S (F_{1}) = S (Q (B)) \cap S (F_{2}) = \emptyset \\ ⟹ A \cap S (F_{1}) = B \cap S (F_{2}) = \emptyset \end{aligned}$
and $\exists$ a homeomorphism $h : (Υ, T \to (Υ, T)$ such that $h (S (F_{1})) \cap S (F_{2}) = \emptyset$ and
$\begin{aligned} h (S (Q (A) \cup F_{1})) = S (Q (B) \cup F_{2}) \\ ⟹ h (A \cup S (F_{1})) = B \cup S (F_{2}) \end{aligned}$
By Theorem 5.4, $h$ is a homeomorphism on $(Υ, T_{0})$ and the proof is done.
Remark 5.8
The above theorem holds only for $T_{0}$ and not for any $T_{a}$ with $a > 0$ as we shall see in the following counter-example.
Example 5.9
Let $0 < b < 1$ be fixed and $Υ = {1, 2, 3, \dots}$ , the set of all natural numbers. Let $T$ be the smallest fuzzy topology generated by the set ${0, 1, 1_{a}, 2_{a}, \dots}$ where $a = \frac{b}{4}$ . Then $T_{b} = {Υ, \emptyset}$ , which is not $A C D H$ . But $(Υ, T)$ is $A F C D H$ since any countable dense $K$ will have its $S (K) = Υ$ .
6. Conclusion

Homogeneity plays a significant role in the study of general topology, as evidenced in Cinderella and Vinod Kumar (2024), Dobrowolski et al. (2021), Fitzpatrick and Hao-xuan (1992), Kennedy (1977, 1990), Ungar (1978), and Van Mill (1982). In this work, several forms of homogeneity namely $n$ -homogeneity, almost countable dense homogeneity, strongly locally homogeneity, and transposition homogeneity have been extended to fuzzy topological spaces and relationships among these fuzzy concepts are well studied. The following are some of the main results of the study.

Fuzzy $n$ -homogeneity, $F S L H, A F C D H, F T H$ are fuzzy topological properties.

$A F C D H (I)$ and $A F C D H (I I)$ are equivalent notions.

If $(Υ, T)$ is a fuzzy homogeneous space containing an open fuzzy crisp point, then $Υ$ is $F S L H$ .

If $(Υ, T)$ is an $S A F C D H, F T H$ space, then $(Υ, T)$ is a $F C D H$ space.

A countable fuzzy separable space $Υ$ is $A F S D$ $⟺$ $Υ$ is $A F C D H$ or $A F C D H^{*}$ .

A countable fuzzy separable space $Υ$ is $A F C D H$ $⟺$ $A F C D H^{*}$ .

If the fuzzy topological space $(Υ, T)$ is $A F C D H$ or $A F S D$ or $F T H$ , so is the corresponding ordinary space $(Υ, T_{0})$ .

In future work, these notions may be extended to intuitionistic fuzzy sets—a generalization of fuzzy sets with wide applications in image processing and pattern recognition.

Footnotes

Acknowledgements

We sincerely thank the reviewers for their valuable comments and the University Grants Commission (UGC) of India for financial support.

ORCID iDs

T J Cinderella

PB Vinod Kumar

Author Contributions

The manuscript was prepared by TJ Cinderella and the PB Vinod Kumar supervised it.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research work of first author is financially supported by the University Grants Commission (UGC), India.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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