Finite Fuzzy Topologies on Boolean Algebras

Abstract

We consider finite B-fuzzy topologies, that is, collections of mappings from a finite set to the Boolean algebra B, satisfying analogous conditions of classical topologies. We establish a one-to-one correspondence between the class of B-fuzzy topologies and the corresponding class of classical ones. Then many consequences follow. For example, we deduce the number of B-fuzzy topologies of small size.

1. Introduction

The concept of fuzzy set theory was initiated by Zadeh [1], who used the closed unit interval $[0,1]$ for the membership set. This was later generalized by Goguen [2] to arbitrary ordered structures, recommending a complete and distributive lattice. Fuzzy topology is a natural extension of general topology, where the latter is a special case. Chang [3] provided the first definition of a fuzzy topology, leading to extensive work by researchers including Gantner, Steinlage, and Warren [4], Schnare [5], Hutton and Reilly [6], Sarkar [7], Badarad and Conard [8,9], Höhle [10,11], and Rodabaugh [12], while Lowen [13,14] modified Chang’s definition to obtain a fuzzy Tychonoff theorem.

Let $X_n=\{1,2,\dots ,n\}$ be an $n-$element set and $B_t$ the finite Boolean algebra of cardinality $2^t$, where t is a positive integer. Let F be the fuzzy set of mappings from $X_n$ into $B_t$ ordered by the usual ordering of mappings, that is,

$$g\le h\iff g\left(i\right)\le h\left(i\right)\mathrm{for}\mathrm{each}i\in X_n,and for any g, h\in F.$$

We write $g<h$ when $g\le h$ and $g\ne h$, i.e., $g\left(i\right)\ne h\left(i\right)$ for some $i\in X_n$. We follow the notations in [15,16,17], and denote an element $g\in F$ by $g=\left(g\right(1),g(2),\dots ,g(n\left)\right)$ or just by $g\left(1\right)g\left(2\right)\dots g\left(n\right)$. The ordered set F is also a Boolean algebra where its least element is $0_F=(0,0,\dots ,0)$ and greatest element is $1_F=(1,1,\dots ,1)$, where 0 and 1 are, respectively, the least and the greatest element of $B_t$. The union ${\displaystyle {\vee }_{i\in I}{g}_i}$ of any collection $\{g_i:i\in I\}$ of elements of F and the intersection $g\wedge h$ of any two elements g and h of F are defined as follows:

$${\displaystyle \underset{i\in I}{\bigvee }{g}_i=\left(\underset{i\in I}{sup}{g}_i\left(1\right),\underset{i\in I}{sup}{g}_i\left(2\right),\dots ,\underset{i\in I}{sup}{g}_i\left(n\right)\right),\mathrm{and}}$$

$$g\wedge h=\left(min\left\{g\right(1),h(1\left)\right\},min\left\{g\right(2),h(2\left)\right\},\dots ,min\left\{g\right(n),h(n\left)\right\}\right).$$

The elements of F will be called $B_t$-fuzzy subsets of $X_n$. The definition of a fuzzy topology is taken from [3].

In this paper, we study finite $B_t$-fuzzy topologies. This work is a generalization of the crisp results in [18,19,20,21], as well as the fuzzy work in [15,16,17], and may be regarded as a continuation of the work in [17], where fuzzy sets with values in the smallest Boolean algebra $D=B_2$ have been considered.

We only consider finite fuzzy topologies on a Boolean algebra, and take the simplest definition of fuzzy topology from [2], which is a generalization of Chang’s definition form [3], namely, replacing the interval $I=[0,1]$ by a distributive lattice L. Similarly, as for the particular case of B-fuzzy topology, the definition of an $L-$topology on any complete bounded lattice L can be given with the appropriate changes. Moreover, it is possible that our ideas regarding computations of finite fuzzy topologies may be generalized to any L-fuzzy topology on more general lattices.

For more information on L-fuzzy topological spaces and some of their recent developments, see [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38], etc., and the references therein. Some references, such as [30,31,32,33,34,35,39], are some recent extensions of L-topologies, and may interest some readers.

Since we used a Chang–Goguen fuzzy topology (the open sets are fuzzy, but the topology comprising those open sets is a crisp subset of the powerset $B_t^L$), the notion of stratified sets and L-topologies, as considered in [40], could be explored in forthcoming research, once the requisite structural complexity is more tractably formalized.

The motivation of this work was facilitated, in part, by the interest in finite topology (fuzzy or crisp) and its applications in the real world, and by the enumeration of finite structures. The importance of this topic resides in the rarity of papers dealing with finite fuzzy structures. Indeed, there are few papers dealing with the subject in the current literature, especially those that use the definition of Goguen, and this paper may be a continuation of such combinatorial investigations that started with [15]. For more details, see the introductions of the papers by [15,16,17].

Fuzzy topologies have many applications in real-life problems, and they enlighten fundamental research in combinatorics and combinatorial structures. Fuzzy topological concepts, such as fuzzy neighborhoods and separation axioms, are foundational to the learning of common optimization techniques and games under uncertainty [41]. The structure of finite fuzzy spaces can be used to model and analyze preferences, risks, and choices in multi-criteria decision-making problems, especially when the set of alternatives is finite [42]. In digital image analysis, the concepts of fuzzy connectedness, fuzzy boundaries, and fuzzy neighborhoods are essential for segmenting and classifying patterns [43,44]. Finite topological structures simplify the computational complexity of these models [45]. The study of fuzzy topological spaces on fuzzy graphs connects topology with graph theory, which has applications in networking problems, computer technology, and data analysis [46]. Fuzzy finite topologies, particularly those generated by fuzzy relations, provide the algebraic and structural basis for analyzing the properties, representations, and characterizations of fuzzy ideals and fuzzy filters, which in turn are used to unify notions of sequence convergence and limits in applied data structures [47]. The Modeling of Uncertain Systems framework is used to model systems where crisp definitions are impossible. For example, the generalized theory of optimal control has used the machinery of fuzzy topological spaces, extending its reach to dynamic systems with imprecise parameters [48].

Our paper is organized as follows: In Section 2, we prove the main result of this paper, that is, the class of finite $B_t$-fuzzy topologies on n points is in a bijective correspondence with the class of the classical topologies on $tn$ points. The number of $B_t$-fuzzy topologies with a small number of open sets, as well as some $B_t$-fuzzy non-discrete topologies with a large number of open sets, will be a direct consequence of the main theorem. This is presented in Section 3 and Section 4. In Section 5, we compute the total number of $B_t$-fuzzy topologies of small size, which is the last concrete consequence of the main result of this paper. The conclusion is provided in Section 6.

2. Main Result

Firstly, some notations and definitions are needed to prove the main result.

Let $t\ge 1$ be a positive integer. We denote by $B_t$ the Boolean algebra with $2^t$ elements, i.e., ${\{0,1\}}^t$ ordered by the following partial order:

For $x,y\in B_t$, $x<y$ if $x_i\le y_i$ for any $i=1,\dots ,t$, and $x_{i_0}<y_{i_0}$ for some $i_0\in \{1,\dots ,t\}$, where $z=(z_1,\dots ,z_t)\in B_t$.

We can extend this partial order to a lexicographical total order as follows: for any $x,y\in B_t$, $x<y$ if $x_1<y_1$, or $x_1=y_1$ and $x_2<y_2$, or …, or $x_i=y_i$, $1\le i\le t-1$, and $x_t<y_t$.

Since any member of the set ${\tau }_t$ of $B_t$-fuzzy subsets is a mapping from an n-set $Y_n=\{y_1,\dots ,y_n\}$ to $B_t$, the fuzzy topology ${\tau }_t$ can be represented by a matrix where the columns are the elements of $Y_n$ and the rows are the fuzzy subsets of ${\tau }_t$. Up to permutations of rows and columns, this matrix is unique. Among all these matrices, we choose the following (canonical) one:

(1)

Fix a total order on $Y_n$: $y_i<y_j$ if $i<j$.

(2)

The columns are ordered according to this order.

(3)

Fix a (lexicographical) total order on $B_t$ that keeps the partial order of $B_t$, as explained above.

(4)

These two total orders imply a partial order on ${\tau }_t$ that we can extend to a lexicographical total order, as performed for $B_t$, since the rows have $\{0,1\}$ entries.

(5)

The rows are ordered according to this total order on ${\tau }_t$.

The obtained matrix is denoted by $M\left({\tau }_t\right)$ and is unique up to the total orders mentioned in (1), (3), and (4). For the remainder of this paper, these orders are fixed for all the considered $B_t$-fuzzy subsets on $Y_n$.

In this case, a classical finite subset is actually a $B_1$-fuzzy subset, i.e., a mapping from a finite set to $\{0,1\}$. Thus, a collection of classical subsets can be represented uniquely by a (canonical) matrix.

For an insight of these definitions and the main proof, readers are recommended to check the example below after Theorem 1.

Let E and A be two finite sets and $C\subseteq A$, t, k, and n be three positive integers. We use the following notations:

$E^A$ or $E^{\left|A\right|}$ are the collection of mappings from A to E.

For any $f\in E^A$, $f\left(C\right)$ is the set all $f\left(y\right)$ for all $y\in C$.

For any $A\subseteq B_t$, and $\alpha \in \{0,1\}$, $f^{\left(\alpha \right)}\in E^A$ such that $f^{\left(\alpha \right)}\left(y\right)=\alpha$ for any $y\in A$, that is, an $\alpha$-constant mapping.

The set $\mathcal{T}\left(n\right)$ is the set of finite (labeled) topologies on n points, and $T\left(n\right)=\left|\mathcal{T}\right(n\left)\right|$.

The set $\mathcal{T}(n,2^t)$ is the corresponding set of finite $B_t$-fuzzy topologies and $T(n,2^t)$ is its cardinality.

The set $\mathcal{T}(n,k)$ is the set of finite (labeled) topologies on n points and having k open sets, and $T(n,k)=\left|\mathcal{T}\right(n,k\left)\right|$ is its cardinality.

The set $\mathcal{T}(n,2^t,k)$ is the corresponding set of finite $B_t$-fuzzy topologies, and $T(n,2^t,k)$ is its cardinality.

Lemma 1. 

Let $t,k,n$ be three positive integers, $X_{tn}=\{x_{11},\dots ,x_{1t},\dots ,x_{tn}\}$, and $Y_n=\{y_1,\dots ,y_n\}$. Then:

(i)

There is a one-to-one correspondence φ between ${\{0,1\}}^{X_{tn}}$ and $B_t^{Y_n}$.

(ii)

$\phi (\varnothing )=0_F$, and $\phi \left(X_{tn}\right)=1_F$, that is, the images of the 0-constant and 1-constant mappings in ${\{0,1\}}^{X_{tn}}$, respectively, are the $0_F$ and $1_F$ mappings in $B_t^{Y_n}$, respectively.

(iii)

For any $\tau \subseteq {\{0,1\}}^{X_{tn}}$, $\left|\tau \right|=\left|\phi \right(\tau \left)\right|$.

Proof. 

For any mapping ${\mu }_{tn}$ from $X_n$ to $\{0,1\}$, its image through $\phi$ is ${\mu }_n$, where ${\mu }_n\left(y_i\right)=({\mu }_{tn}\left(x_{i1}\right),\dots ,{\mu }_{tn}\left(x_{it}\right))\in {\{0,1\}}^t=B_t$ for any $y_i\in Y_n$. It is clear that $\phi$ is injective. It is also surjective because ${\phi }^{-1}\left({\mu }_n\right)\left(x_{ij}\right)$ is the jth component of ${\mu }_n\left(y_i\right)$, and thus ${\phi }^{-1}\left({\mu }_n\right)\in {\{0,1\}}^{X_{tn}}$. This means that $\phi$ is bijective. The second part of the lemma is straightforward. For the third part, since $\tau$ and $\phi \left(\tau \right)$ are represented by the two identical matrices $M\left(\tau \right)$ and $M\left(\phi \right(\tau \left)\right)$ under the above conditions (1)–(5), $\left|\tau \right|=$ number of rows of $M\left(\tau \right)=$ number of rows of $M\left(\phi \right(\tau \left)\right)=\left|\phi \right(\tau \left)\right|$. □

Below, we prove the main result of this paper.

Theorem 1. 

Let $t,k,n$ be three positive integers. Then there is a one-to-one correspondence between $\mathcal{T}(tn,k)$ and $\mathcal{T}(n,B_t,k)$.

Proof. 

It suffices to prove that the one-to-one correspondence $\phi$ introduced in the proof of the above lemma induces a one-to-one correspondence between the considered sets.

• Let ${\mu }_{tn,i}$ be two mappings from $X_{tn}$ to $\{0,1\}$, $i=1,2$ (in other words, two subsets of $X_{tn}$), ${\mu }_{n,i}=\phi \left({\mu }_{tn,i}\right)$, ${\mu }_{tn}={\mu }_{tn,1}\vee {\mu }_{tn,2}$, and ${\mu }_n={\mu }_{n,1}\vee {\mu }_{n,2}$. We can now prove that $\phi \left({\mu }_{tn}\right)={\mu }_n$.
• Let $\lambda =\phi \left({\mu }_{tn}\right)\in B_t^{Y_n}$, and $y_i\in Y_n$. It follows that

  $$\lambda \left(y_i\right)=({\mu }_{tn}\left(x_{i1}\right),\dots ,{\mu }_{tn}\left(x_{it}\right))=(({\mu }_{tn,1}\vee {\mu }_{tn,2})\left(x_{i1}\right),\dots ,({\mu }_{tn,1}\vee {\mu }_{tn,2})\left(x_{it}\right))$$

  $$=(max\{{\mu }_{tn,1}\left(x_{i1}\right),{\mu }_{tn,2}\left(x_{i1}\right)\},\dots ,max\{{\mu }_{tn,1}\left(x_{it}\right),{\mu }_{tn,2}\left(x_{it}\right)\}).$$
• On the other hand,

  $${\mu }_n\left(y_i\right)=({\mu }_{n,1}\vee {\mu }_{n,2})\left(y_i\right)=max\{{\mu }_{t,1}\left(y_i\right),{\mu }_{n,2}\left(y_i\right)\}$$

  $$=max\left\{\right({\mu }_{tn,1}\left(x_{i1}\right),\dots ,{\mu }_{tn,1}\left(x_{it}\right),({\mu }_{tn,2}\left(x_{i1}\right),\dots ,{\mu }_{tn,2}\left(x_{it}\right)\}(*)$$

  $$=(max\{{\mu }_{tn,1}\left(x_{i1}\right),{\mu }_{tn,2}\left(x_{i1}\right)\},\dots ,max\{{\mu }_{tn,1}\left(x_{it}\right),{\mu }_{tn,2}\left(x_{it}\right)\}),$$

  because the two vectors in (*) belong to $B_t$ and can be viewed as mappings from $\{x_{i1},\dots ,x_{it}\}$ to $\{0,1\}$. This the desired result.

The corresponding equality for the intersection is similarly proved.

Finally, if ${\mu }_{tn,1}\le {\mu }_{tn,2}$, then ${\mu }_{tn,1}\left(x_{ij}\right)\le {\mu }_{tn,2}\left(x_{ij}\right)$ for all $x_{ij}\in X_{tn}$. This yields ${\mu }_{n,1}\left(y_i\right)\le {\mu }_{n,2}\left(y_i\right)$ for all $y_i\in Y_n$. In other words, ${\mu }_{n,1}\le {\mu }_{n,2}$, i.e., $\phi$ keeps the order.

Assertion (iii) of Lemma 1 implies that the number of open sets of any classical topology $\tau$ on $X_{tn}$ is the same number of $B_t$-fuzzy open sets of its image $\phi \left(\tau \right)$, the $B_t$-fuzzy topology defined on $Y_n$, and vice versa. □

To illustrate this result and the tools used to achieve it, we provide an example below.

Example 1. 

$n=3$, and $t=2$.

Consider the Boolean algebra $B_2=\{0_B,a,b,1_B\}$, where $0<a,b<1$, $X_6=\{x_{11},x_{12},x_{21},x_{22},x_{31},x_{32}\}$ and $Y_3=\{y_1,y_2,y_3\}$. Now, let ${\tau }_B=\{f_i$, $1\le i\le 7\}$ be the $B_2$-fuzzy topology on $Y_3$, where the mappings $f_i$ range from $Y_3$ to $B_2$, as summarized in

Table 1. Matrices representing ${\tau }_B$ (on the left) and its corresponding classical topology $\tau$ (on the right).

	${\mathit{y}}_1$	${\mathit{y}}_2$	${\mathit{y}}_3$		${\mathit{x}}_{11}$	${\mathit{x}}_{12}$	${\mathit{x}}_{21}$	${\mathit{x}}_{22}$	${\mathit{x}}_{31}$	${\mathit{x}}_{32}$
$f_0$	$0_B$	$0_B$	$0_B$	∅	0	0	0	0	0	0
$f_2$	b	$0_B$	$0_B$	$\left\{x_{11}\right\}$	1	0	0	0	0	0
$f_3$	a	$0_B$	$0_B$	$\left\{x_{12}\right\}$	0	1	0	0	0	0
$f_4$	$1_B$	$0_B$	$0_B$	$\{x_{11},x_{12}\}$	1	1	0	0	0	0
$f_5$	$1_B$	b	$0_B$	$\{x_{11},x_{12},x_{21}\}$	1	1	1	0	0	0
$f_6$	$1_B$	a	$0_B$	$\{x_{11},x_{12},x_{22}\}$	1	1	0	1	0	0
$f_1$	$1_B$	$1_B$	$1_B$	$X_6$	1	1	1	1	1	1

. For example, $f_5$ is the mapping from $Y_3$ to $B_2$, such that $f_5\left(y_1\right)=1_B$, $f_5\left(y_2\right)=b$, $f_5\left(y_3\right)=0_B$, while $\tau =\{\varnothing ,\left\{x_{11}\right\},\left\{x_{12}\right\},\{x_{11},x_{12}\},\{x_{11},x_{12},x_{21}\},\{x_{11},x_{12},x_{22}\},X_6\}$ is a classical topology defined on $X_6$ and represented by a matrix in Table 1. For example, the corresponding row for the open set $\{x_{11},x_{12},x_{22}\}$ is $[1,1,0,1,0,0]$, giving information on which point $x\in X_6$ belongs to this open set and which one does not.

By using the binary representation of $B_2$ as ${\{0,1\}}^2=\{(0,0),(0,1),(1,0),(1,1)\}$, where we identify $0_B$ with $(0,0)$, a with $(0,1)$, b with $(1,0)$, and finally $1_B$ with $(1,1)$, we obtain the corresponding classical topology $\tau$ on $X_6$ from ${\tau }_B$ and vice versa. For example, the blue entries of the matrix representing ${\tau }_B$ correspond to the blue entries of the matrix repersenting $\tau$. And so on for the black and green entries of both matrices. As the reader can see, the number of open sets is the same, i.e., seven open sets. This also illustrates Corollary 1.

Our first two direct consequences of the main result are as follows:

Corollary 1. 

$T(tn,k)=T(n,2^t,k)$.

Corollary 2. 

$T\left(tn\right)=T(n,2^t)$.

3. $\mathit{B}-$Fuzzy Topologies with a Small Number of Open Sets

Before providing some other consequences of our main result, let us recall the definition of the Stirling number of the second kind.

Definition 1. 

The number of partitions of a finite set with cardinality $n\ge 1$ into k subsets is called the Stirling number of the second kind, and is denoted as $S_{n,k}$.

By the principle of inclusion-exclusion principle, one can prove that $S_{n,k}$ is

$$S_{n,k}={\displaystyle \frac{1}{k!}}\sum _{j=0}^k{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){(k-j)}^n$$

In [18], the number of classical topologies on $X_n$ with k open sets, $4\le k\le 12$, was computed. Using Theorem 1 below, we deduce the third consequence of our main result, that is, the number of fuzzy topologies with k open sets. Actually, this is an application of Corollary 1, which gives the number of B-fuzzy topologies by means of their corresponding classical ones.

Theorem 2. 

The number $T(n,2^t,k)$ of $B_t-$ fuzzy topologies of cardinality k, $4\le k\le 12$ is given by

$$\begin{array}{cc}\hfill T(n,2^t,4)=& 3^{tn}-3\left(2^{tn}\right)+3\hfill \\ \hfill T(n,2^t,5)=& 4^{tn}-3^{tn+1}+{3.2}^{tn}-1\hfill \\ \hfill T(n,2^t,6)=& 3!S_{nt,3}+{\displaystyle \frac{3}{2}}4!S_{nt,4}+5!S_{nt,5}\hfill \\ \hfill T(n,2^t,7)=& {\displaystyle \frac{9}{4}}4!S_{nt,4}+2.5!S_{nt,5}+6!S_{nt,6}\hfill \\ \hfill T(n,2^t,8)=& S_{nt,3}+2.4!S_{nt,4}+{\displaystyle \frac{15}{4}}5!S_{nt,5}+{\displaystyle \frac{15}{2}}6!S_{nt,6}+7!S_{nt,7}\hfill \\ \hfill T(n,2^t,9)=& {\displaystyle \frac{5}{6}}4!S_{nt,4}+5.5!S_{nt,5}+{\displaystyle \frac{11}{2}}6!S_{nt,6}+3.7!S_{nt,7}+8!S_{nt,8}\hfill \\ \hfill T(n,2^t,10)=& 4.4!S_{nt,4}+{\displaystyle \frac{11}{2}}.5!S_{nt,5}+{\displaystyle \frac{73}{8}}6!S_{nt,6}+{\displaystyle \frac{15}{2}}.7!S_{nt,7}+{\displaystyle \frac{7}{2}}8!S_{nt,8}+9!S_{nt,9}\hfill \\ \hfill T(n,2^t,11)=& {\displaystyle \frac{11}{2}}.5!S_{nt,5}+{\displaystyle \frac{73}{8}}6!S_{nt,6}+{\displaystyle \frac{15}{2}}.7!S_{nt,7}+{\displaystyle \frac{7}{2}}8!S_{nt,8}+9!S_{nt,9}\hfill \\ \hfill T(n,2^t,12)=& {\displaystyle \frac{1}{2}}{S}_{nt,4}+{\displaystyle \frac{11}{2}}.5!S_{nt,5}+{\displaystyle \frac{73}{8}}6!S_{nt,6}+{\displaystyle \frac{15}{2}}.7!S_{nt,7}+{\displaystyle \frac{7}{2}}8!S_{nt,8}+9!S_{nt,9}+10!S_{nt,10}+11!S_{nt,11}\hfill \end{array}$$

4. $\mathit{B}-$Fuzzy Topologies with Large Number of Open Sets

As for the classical case, the total number of topologies with a large number of open sets is known, so we can provide its analogy in the $B_t-$ fuzzy case. Corollary 2 implies that the number of $B_t$-fuzzy topologies on n points is equal to the number of classical ones on $tn$ points. So, the fourth consequence of our main result is as follows:

Theorem 3. 

There are no $B_t$-fuzzy topology of cardinality k on $Y_n$, where k is an integer $3\times 2^{tn-2}<k<2^{tn}.$

Proof. 

This is an immediate consequence from the classical case, since there are no topologies of cardinality k on $X_n$, with $3\times 2^{n-2}<k<2^n.$ □

The fifth consequence provides the total number of $B_t$-topologies in the previous extreme case, as follows:

Theorem 4. 

The total number of $B_t$-topology with cardinality $3\times 2^{nt-2}$ is given by $nt(nt-1)$.

Proof. 

This is also immediate from the classical case, since the number of topologies of cardinality $3\times 2^{n-2}$ is $n(n-1)$. □

Moreover, all results proved in [17] become corollaries of our main result, and are direct consequences of it.

5. Number of $\mathbf{B}-$Fuzzy Topologies for Small $\mathbf{n}$ and $\mathbf{t}$

The sixth consequence is the following direct implication of Corollary 2.

Theorem 5. 

The number $T(n,B_t)$ of all labeled $B_t-$ fuzzy topologies on $Y_n$ for small n and t is given by the following:

$$\begin{array}{cc}\hfill T(2,B_2)=& 355\hfill \\ \hfill T(2,B_3)=& T(3,B_2)=209,527\hfill \\ \hfill T(2,B_4)=& T(4,B_2)=642,779,354\hfill \\ \hfill T(3,B_3)=& 63,260,289,423\hfill \\ \hfill T(2,B_5)=& T(5,B_2)=8,977,053,873,043\hfill \\ \hfill T(2,B_6)=& T(3,B_4)=T(4,B_3)=519,355,571,065,774,021\hfill \\ \hfill T(2,B_7)=& T(7,B_2)=115,617,051,977,054,267,807,460\hfill \\ \hfill T(3,B_5)=& T(5,B_3)=88,736,269,118,586,244,492,485,121\hfill \\ \hfill T(2,B_8)=& T(8,B_2)=T(4,B_4)=93,411,113,411,710,039,565,210,494,095\hfill \\ \hfill T(2,B_9)=& T(9,B_2)=T(3,B_6)=T(6,B_3)=261,492,535,743,634,374,805,066,126,901,117,203\hfill \end{array}$$

and so on. We can obtain as many as consequences as necessary, and then obtain appropriate results for classical topologies.

6. Conclusions

The primary achievement of the current research is the establishment of a direct link between B-fuzzy topologies and their classical topological counterparts, a key result that provides a framework for translating concepts between the two theories. Building upon this success, the authors propose to extend this investigation by generalizing the lattice structure used to define the topologies. Specifically, future work should focus on P-fuzzy topologies, where the fuzziness is valued in P, an arbitrary bounded poset (partially ordered set). The ultimate goal is to examine whether the relationship with classical topologies can be maintained or generalized under this less restrictive structure. The authors acknowledge that initial progress on this generalization has already been made in [16], but only for special cases where P is a totally ordered poset (a chain). This confirms that the direction of future research should be to move from specialized fuzzy topological spaces to those defined using increasingly general algebraic structures. Another direction is to prove results on B-fuzzy topologies that are not known for the classical case. We can then deduce the analogies for finite topologies and obtain new results in the classical case.