Topological Spaces on Fuzzy Structures

MICHAEL GR. VOSKOGLOU

Mathematical Sciences, School of Technological Applications

University of Peloponnese (ex T.E.I. of Western Greece)

Meg. Alexandrou 1, 26334 Patras

GREECE

Abstract: - Apart from their applications to almost all sectors of the human activity, fuzzy mathematics is also

importantly developed on a theoretical basis providing useful links even to classical branches of pure

mathematics, like Algebra, Analysis, Geometry, Topology, etc. The present paper comes across the steps that

enabled the extension of the concept of topological space, the most general category of mathematical spaces, to

fuzzy structures. Fuzzy and soft topological spaces are introduced in particular, the fundamental concepts of

limits, continuity, compactness and Hausdorff space are defined on them and examples are provided illustrating

them.

Key-Words: - Fuzzy set, soft set, fuzzy topological space (FTS), soft topological space (STS).

Received: July 20, 2021. Revised: June 21, 2022. Accepted: July 25, 2022. Published: September 1, 2022.

1 Introduction

Since Zadeh introduced the fuzzy set theory in 1965

[1], a lot of research was carried out for improving

its effectiveness to deal with uncertain, ambiguous

and vague situations. As a result a series of

extensions and generalizations of the concept of

fuzzy set have been reported (e.g. interval-valued

fuzzy set, type-2 fuzzy set, intuitionistic fuzzy set,

neutrosophic set, etc.) and several theories have

been proposed (e.g. rough sets, soft sets, grey

systems, etc.) as alternatives to the fuzzy set theory

(e.g. see [2]).

Those new mathematical tools gave to experts the

opportunity to model under conditions which are

vague or not precisely defined, thus succeeding to

solve mathematically problems whose statements

are expressed in our natural language without exact

numerical data. As a consequence, the spectre of

applications of fuzzy sets and of the related to them

extensions/theories has been rapidly extended

covering nowadays almost all sectors of the human

activity (Physical Sciences, Economics and

Management, Expert Systems, Industry, Robotics,

Decision Making, Programming, Medicine,

Biology, Humanities, Human Reasoning, Education,

etc.); e.g. see [3, Chapter 6], [4-7], etc.

Fuzzy mathematics, however, has been also

importantly developed on a theoretical basis

providing useful links even to classical branches of

pure mathematics, like Algebra, Analysis,

Geometry, Topology, etc.

Topological spaces [8], for instance, is the most

general category of mathematical spaces, where

fundamental concepts like limits, continuity,

compactness, etc., are defined. The present paper

comes across the steps that enabled the extension of

topological spaces to fuzzy structures. The concepts

of fuzzy topological space (FTS) and of soft

topological spaces (STS) are introduced in

particular, and examples are presented illustrating

these concepts. More explicitly, FTSs are defined in

Section 2, STSs are defined in Section 3 and the

article closes with the final conclusions and some

hints for future research, contained in its last Section

2 Fuzzy Topological Spaces

2.1 Fuzzy Sets

Zadeh introduced the concept of fuzzy set as follows

[1]:

Definition 1: A fuzzy set Α in the universe U is

defined with the help of its membership function m:



[0,1] as the set of the ordered pairs

A = {(x, m(x)): x



U} (1)

The real number m(x) is called the membership

degree of x in Α. The greater m(x), the more x

satisfies the characteristic property of Α. Many

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.72

Michael Gr. Voskoglou

E-ISSN: 2224-2880

624

Volume 21, 2022

authors, for reasons of simplicity, identify a fuzzy

set with its membership function.

There is a difficulty, however, with the definition of

the membership function, which is not unique,

depending on the “signals” that each one receives

from the real word. An individual of height 1.80 m,

for example, may be considered as being “tall” by

one observer and as having a regular height (not

“tall”) by another one. The methods used for

defining the membership functions are usually

statistical or empirical. The only restriction

concerning the definition of a membership function

is to be compatible with common logic; otherwise

the fuzzy set does not give a reliable representation

of the corresponding real situation. This could

happen, for instance, if individuals of height ≤1.50

m possess membership degrees ≥ 0.5 in the fuzzy

set of “tall people”.

A crisp subset A of U is a fuzzy set in U with

membership function taking the values m(x)=1, if x

belongs to A, and 0 otherwise.

The basic definitions of crisp sets are generalized in

a natural way to fuzzy sets as follows [3]:

Definition 2: The universal fuzzy set FU and the

empty fuzzy set F∅ in the universe U are defined as

the fuzzy sets in U with membership functions

m(x)=1 and m(x)=0 respectively, for all x in U.

It is straightforward to check that for each fuzzy set

A in U is A∪FU = FU, A∩FU = A, A∪F∅ = A and

A∩F∅ = F∅.

Definition 3: Let A and B be two fuzzy sets in the

universe U with membership functions mA and mB

respectively. Then A is said to be a fuzzy subset of

B, if mA(x)≤ mB(x), for all x in U. We write then A



B. If mA(x)< mB(x), for all x in U, then A is

called a proper fuzzy subset of B and we write A



Definition 4: Let A and B be two fuzzy sets in the

universe U with membership functions mA and mB

respectively. Then:

 The union A∪B is defined to be the FS in U

with membership function mA∪B(x)=max

{mA(x), mB (x)}, for each x in U.

 The intersection A∩B is defined to be the

FS in U with membership function

mA∩B(x)=min {mA(x), mB (x)}, for each x in

 The complement of A is defined as the FS

AC in U with membership function mC(x) =

1- m(x), for all x in U.

Example 1: Let U be the set of the human ages

U={5,10,20,30,40,50,60,70,80}. Table 1 gives the

membership degrees of the elements of U with

respect to the fuzzy sets A=young, B=adult and

C=old in U.

1. Prove that

CB

. Is

CB

2. Calculate the fuzzy sets

AC

and

(A C) B

Table 1: Human ages

0.8

0.5

0.2

0.4

0.1

0.6

0.8

Solution: 1) From Table 1 turns out that

m (x) (x), x U   

. Thus

CB

. But

m (70) m (70) 1

, therefore it is not true that

CB

2) Calculating min

 

m (x), m (x) , x U

one

finds that A∩C = {(5, 0), (10, 0), (20, 0), (30, 0.2),

(40, 0.2), (50, 0.1), (60, 0), (70, 0), (80, 0)}.

Also, calculating max

 

A C B

m (x), m (x) ,



∀ x ∈

U, one finds that

(A C) B

= {(5, 0), (10, 0), (20,

0), (30, 0.2), (40, 1), (50, 1), (60, 1), (70, 1), (80,

1)}.

2.2 Fuzzy Topologies

Definition 5 [9]: A fuzzy topology (FT) T on a non-

empty set U is a collection of FSs in U such that:

 The universal and the empty FSs belong to

 The intersection of any two elements of T

and the union of an arbitrary (finite or

infinite) number of elements of T belong also

to T.

Examples 2: Trivial examples of FTs are the

discrete FT {F∅, FU} and the non-discrete FT

consisting of all FSs in U. Another example is the

collection of all constant FSs in U, i.e. all FSs in U

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.72

Michael Gr. Voskoglou

E-ISSN: 2224-2880

625

Volume 21, 2022

with membership function of the form m(x)=c, for

some c in [0, 1] and all x in U.

The elements of a FT T on U are called open fuzzy

sets in U and their complements are called closed

fuzzy sets in U. The pair (U, T) defines a fuzzy

topological space (FTS) on U.

Next we describe how one can extend the concepts

of limit, continuity, compactness, and Hausdorff

space to FTSs [9].

Definition 6: Given two fuzzy sets A and B of the

FTS (U, T), B is said to be a neighborhood of A, if

there exists an open fuzzy set O such that A





Definition 7: We say that a sequence {An} of fuzzy

sets of (U, T) converges to the fuzzy set A of (U, T),

if there exists a positive integer m, such that for

each integer n≥m and each neighborhood B of A we

have that An



B. Then A is called the limit of {An}.

Lemma 1: (Zadeh’s extension principle) Let X and

Y be two non-empty crisp sets and let f: X



Y be a

function. Then f can be extended to a function F

mapping FSs in X to fuzzy sets in Y.

Proof: Let A be a fuzzy set in X with membership

function mA. Then its image F(A) is the fuzzy set B

in Y with membership function mB, which is defined

as follows: Given y in Y, consider the set f -1(y)={x

∈ X: f(x)=y}. If f -1(y)=∅, then mB(y)=0, and if

f-1(y)≠∅, then mB(y) is equal to the maximal value of

all mA(x) such that x ∈ f -1(y). Conversely, the

inverse image F-1(B) is the fuzzy set A in X with

membership function mA(x)=mB(f(x)), for each x ∈

Definition 8: Let (X, T) and (Y, S) be two FTSs and

let f: X



Y be a function. By Lemma 1, f can be

extended to a function F mapping fuzzy sets in X to

fuzzy sets in Y. We say then that f is a fuzzy

continuous function, if, and only if, the inverse

image of each open fuzzy set in Y through F is an

open fuzzy set in X.

Definition 9: A family A={Ai, i∈I} of fuzzy sets of

a FTS (U, T) is called a cover of U, if U=



. If

the elements of A are open fuzzy sets, then A is

called an open cover of U. Also, each subset of A

being also a cover of U is called a sub-cover of A.

The FTS (U, T) is said to be compact, if every open

cover of U contains a sub-cover with finitely many

elements.

Definition 10: A FTS (U, T) is said to be:

1. A T1-FTS, if, and only if, for each pair of

elements u1, u2 of U, u1≠u2, there exist at

least two open fuzzy sets O1 and O2 such

that u1∈O1, u2



O1 and u2∈O2, u1



O2.

2. A T2-FTS (or a separable or a Hausdorff

FTS), if, and only if, for each pair of

elements u1, u2 of U, u1≠ u2, there exist at

least two open fuzzy sets O1 and O2 such

that u1∈O1, u2∈O2 and O1∩O2 = ∅F.

Obviously a T2-FTS is always a T1-FTS.

3 Soft Topological Spaces

3.1 Soft Sets

The need of passing through the existing difficulty

to define properly the membership function of a

fuzzy set gave the hint to D. Molodstov, Professor

of Mathematics at the Russian Academy of

Sciences, to introduce in 1999 the concept of soft

set [10] as a tool to tackle the existing in real world

uncertainty in a parametric manner. A soft set is

defined as follows:

Definition 11: Let E be a set of parameters, let A be

a subset of E, and let f be a map from A into the

power set P(U) of all subsets of the universe U.

Then the soft set (f, A) in U is defined to be the set

of the ordered pairs

(f, A) = {(e, f(e)): e ∈ A} (2)

In other words, a soft set in U is a parametrised

family of subsets of U. The name "soft" was given

because the form of (f, A) depends on the

parameters of A. For each e ∈ A, its image f(e) is

called the value set of e in (f, A), while f is called

the approximation function of (f, A).

Example 3: Let U= {C1, C2, C3} be a set of cars and

let E = {e1, e2, e3} be the set of the parameters

e1=cheap, e2=hybrid (petrol and electric power) and

e3= expensive. Let us further assume that C1, C2 are

cheap, C3 is expensive and C2, C3 are the hybrid

cars. Then, a map f: E



P(U) is defined by

f(e1)={C1, C2}, f(e2)={C2, C3} and f(e3)={C3} .

Therefore, the soft set (f, E) in U is the set of the

ordered pairs (f, E) = {(e1, {C1, C2}), (e2, {C2, C3},

(e3, {C3}}.

A fuzzy set in U with membership function y = m(x)

is a soft set in U of the form (f, [0, 1]), where

f(α)={x



U: m(x)



α} is the corresponding α – cut

of the fuzzy set, for each α in [0, 1].

It is of worth noting that, apart from soft sets, which

overpass the existing difficulty of defining properly

membership functions through the use of the

parameters of E, alternative theories for managing

the uncertainty have been also developed, where the

definition of a membership function is either not

necessary (grey systems/numbers [11]), or it is

overpassed by using a pair of crisp sets which

give the lower and the upper approximation of

the original crisp set (rough sets [12]).

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.72

Michael Gr. Voskoglou

E-ISSN: 2224-2880

626

Volume 21, 2022

The basic definitions on soft sets are introduced in a

way analogous to fuzzy sets (see section 2.1)

Definition 12: The absolute soft set AU is defined to

be the soft set (f, A) such that f(e)=U, ∀ e ∈ A, and

the null soft set A∅ is defined to be the soft set (f, A)

such f(e)=∅, ∀ e ∈A.

It is straightforward to check that for each soft set A

in U is A∪AU = AU, A∩ AU = A, A∪ A∅ = A and

A∩ A∅ = A∅.

Definition 13: If (f, A) and (g, B) are two soft sets

in U, (f, A) is called a soft subset of (g, B), if A



and f(e)



g(e), ∀ e ∈A. We write then (f, A)



(g,

B). If A



B, then (f, A) is called a proper soft

subset of B and we write (f, A)



(g, B).

Definition 14: Let (f, A) and (g, B) be two soft sets

in U. Then:

 The union (f, A) ∪ (g, B) is the SS (h, A∪B)

in U, with h(e)=f(e) if e ∈ A-B, h(e)=g(e) if

e ∈ B-A and h(e)= f(e)∪g(e) if e ∈ A∩B.

 The intersection (f, A) ∩ (g, B) is the soft

set (h, A∩B) in U, with h(e)=f(e)∩g(e), ∀ e

∈ A∩B.

 The complement (f, A)C of the soft SS (f, A)

in U, is the SS (fC, A) in U, for which the

function fC is defined by fC(e) = U-f(e), ∀ e

∈ A.

For general facts on soft sets we refer to [13].

Example 4: Let U={H1, H2, H3}, E={e1, e2, e3} and

A= {e1, e2}. Consider the soft set

S = (f, A) = {(e1, {H1, H2}), (e2, {H2, H3})} in U.

Then the soft subsets of S are the following:

S1={(e1, {H1})}, S2={(e1, {H2})}, S3={(e1, {H1,

H2})}, S4={(e2, {H2})}, S5={(e2, {H3})}, S6={(e2,

{H2, H3})}, S7={(e1, {H1}, (e2, {H2})}, S8={(e1,

{H1}, (e2, {H3})}, S9={(e1, {H2}, (e2, {H2})},

S10={(e1, {H2}, (e2, {H3})}, S11={(e1, {H1, H2}, (e2,

{H2})}, S12={(e1, {H1, H2}, (e2, {H3})}, S13={(e1,

{H1}, (e2, {H2, H3})}, S14={(e1, {H2}, (e2, {H2,

H3})}, S, A∅={(e1, ∅), (e2, ∅)}

It is also easy to check that (f, A)C= {(e1, {H3}), (e2,

{H1})}.

3.3 Soft Topologies

Observe that the concept of FTS (Definition 5) is

obtained from the classical definition of TS [8] by

replacing in it the expression “a collection of subsets

of U” by the expression “a collection of FSs in U”.

In an analogous way one can define the concepts of

intuitionistic FTS (IFTS) [14], of neutrosophic TS

(NTS) [15, 16], of rough TS (RTS) [17], of soft TS

(STS) [18], etc. In particular, a STS is defined as

follows:

Definition 15: A soft topology T on a non-empty set

U is a collection of SSs in U with respect to a set of

parameters E such that:

 The absolute and the null soft sets EU and

E∅ belong to T

 The intersection of any two elements of T

and the union of an arbitrary (finite or

infinite) number of elements of T belong

also to T.

The elements of a ST T on U are called open SS and

their complements are called closed SS. The triple

(U, T, E) is called a STS on U.

Examples 5: Trivial examples of STs are the

discrete ST {E∅, EU} and the non-discrete ST

consisting of all SSs in U. Reconsider also Example

4. It is straightforward to check then that T = {EU,

E∅, S, S2, S9, S11} is a ST on U.

The concepts of limit, continuity, compactness, and

Hausdorff TS are extended to STSs in a way

analogous of FTSs [19, 20]. In fact, Definitions 7, 9

and 10 are easily turned to corresponding definitions

of STSs by replacing the expression “fuzzy sets”

with the expression “soft sets”. For the concept of

continuity we need the following Lemma ([19],

definition 3.12) :

Lemma 2: Let (U, T. A), (V, S, B) be STSs and let

u: U→V, p: A→B be given maps. Then a map fpu is

defined with respect to u and p mapping the soft sets

of T to soft sets of S.

Proof: If (F, A) is a soft set of T, then its image

fpu((F, A)) is a soft set of S defined by

fpu((F, A))=(fpu(F), p(A)), where, ∀ y ∈ B is

fpu(F)(y)=

-1

x p (y) A

u(F(x))



if p-1(y)∩A≠∅ and

fpu(F)(y) = ∅ otherwise.

Conversely, if (G, B) is a soft set of S, then its

inverse image fpu-1((G, B)) is a soft set of T defined

by fpu-1((G, B))=(fpu-1(G), p-1(B)), where ∀ x ∈ A is

fpu-1(G)(x)=u-1(G(p(x)).

Definition 16: Let (U, T. A), (V, S, B) be STSs and

let u: U→V, p: A→B be given maps. Then the map

fpu, defined by Lemma 2, is said to be soft pu-

continuous, if, and only if, the inverse image of each

open soft set in Y through fpu is an open soft set in

4. Discussion and Conclusions

In this review paper we extended the classical

concept of TS to FTS and STS and we generalized

in them the fundamental notions of limit, continuity,

compactness and Hausdorff space. Further

extensions of the concept of TS to IFTS, NTS and to

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.72

Michael Gr. Voskoglou

E-ISSN: 2224-2880

627

Volume 21, 2022

other fuzzy structures are very important and could

be studied by the author in a future review paper.

In general, apart from its many and important

practical applications (e.g. [21,22], etc.), fuzzy

mathematics has been also significantly developed

on a theoretical basis, providing useful links even to

classical branches of pure mathematics, like

Algebra, Analysis, Geometry, Topology, etc. (e.g.

[23,24], etc.) This is, therefore, a promising area for

future research.

References:

[1] Zadeh, LA., Fuzzy Sets, Information and

Control, 8, 1965, pp. 338-353.

[2] Voskoglou, M.Gr., Fuzzy Systems, Extensions

and Relative Theories, WSEAS Transactions on

Advances in Engineering Education, 16, 2019,

pp.63-69

[3] Klir, G. J. & Folger, T. A., Fuzzy Sets,

Uncertainty and Information, Prentice-Hall,

London, 1988.

[4] Deng, J., Introduction to Grey System Theory,

The Journal of Grey System, 1, 1989, pp. 1-24.

[5] Kharal, A. & Ahmad, B., Mappings on soft

classes, New Mathematics and Natural Computation,

7(3), 2011, pp.471-481.

[6] Voskoglou, M.Gr., Finite Markov Chain and

Fuzzy Logic Assessment Models: Emerging

Research and Opportunities, Create Space

Independent Publishing Platform, Amazon,

Columbia, SC, USA, 2017.

[7] Atanassov, K.T., Intuitionistic Fuzzy Sets,

Physica-Verlag, Heidelberg, N.Y., 1999.

[8] Willard, S., General Topology, Dover Publ. Inc.,

N.Y., 2004.

[9] Chang, S.L., Fuzzy Topological Spaces, Journal

of Mathematical Analysis and Applications, 24(1),

1968, pp. 182-190.

[10] Molodtsov, D. , Soft Set Theory-First Results,

Computers and Mathematics with Applications,

37(4-5), 1999, pp. 19-31.

[12] Deng, J., Control Problems of Grey Systems,

Systems and Control Letters, 1982, pp. 288-294.

[13] Pawlak, Z., Rough Sets: Aspects of Reasoning

about Data, Kluer Academic Publishers, Dordrecht,

1991.

[14] Luplanlez, F.G, On intuitionistic fuzzy

topological spaces, Kybernetes, 35(5), 2006, pp.743-

747.

[15] Salama, A.A., Alblowi, S.A., Neutrosophic Sets

and Neutrosophic Topological Spaces, IOSR Journal

of Mathematics, 3(4), 2013, pp. 31-35.

[16] Voskoglou, M.Gr., Managing the Existing in

Real Life Indeterminacy, Int. J. of Mathematical and

Computational Methods, 7, 2022, pp.29-34.

[17] Lashin, E.F., Kozae, A.M., Abo Khadra, A.A. &

Medhat, T., Rough set for topological spaces,

International J. Approx. Reason. 40, 2005, pp. 35–

43.

[18] Shabir, M. & Naz M. (2011), On Soft

Topological Spaces, Computers and Mathematics

with Applications, 61, 2011, pp. 1786-1799.

[19] Zorlutuna, I., Akdag, M., Min, W.K. & Amaca,

S., Remarks on Soft Topological Spaces, Annals of

Fuzzy Mathematics and Informatics, 3(2), 2011, pp.

171-185.

[20] Georgiou, D.N., Megaritis, A.C., Petropoulos,

V.I., On Soft Topological Spaces, Applied

Mathematics & Information Sciences, 7(5), 2013,

pp. 1889-1901.

[21] Aryandani, A., Nurjannah, S., Achmad, A.

Fernandes, R., Efendi, A., Implementation of Fuzzy

C-Means in Investor Group in the Stock Market

Post-Covid-19 Pandemic, WSEAS Transactions on

Mathematics, 21, 2022, pp. 415-423,.

[22] Voskoglou, M.Gr., A Hybrid Model for

Decision Making Utilizing TFNs and Soft Sets as

Tools, WSEAS Transactions on Equations, 2, 2022,

pp. 65-69.

[23] Sahni, A.K., Pandey, J.T. Mishra, R.K,

Cancellation On Fuzzy Projective Modules And

Schanuel’s Lemma Using Its Conditioned Class,

WSEAS Transactions on Mathematics, 21, 2022,

pp. 476-486.

[24] Al-Ηusban, A., Fuzzy Soft Groups Based on

Fuzzy Space, WSEAS Transactions on Mathematics,

21, 2022, pp. 53-57.

Creative Commons Attribution

License 4.0 (Attribution 4.0

International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.72

Michael Gr. Voskoglou

E-ISSN: 2224-2880

628

Volume 21, 2022