# On FuzzySp-Open Sets

Authors: Hakeem A. Othman
Journal: Advances in Fuzzy Systems
 (2011)

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## Abstract

A new class of generalized fuzzy open sets in fuzzy topological space,
called fuzzysp-open sets, are introduced, and their properties are studied and the relationship between this new concept and other weaker forms of fuzzy open sets we discussed. Moreover, we introduce the fuzzy sp-continuous (resp., fuzzy sp-open) mapping and other stronger forms of sp-continuous (resp., fuzzy sp-open) mapping and establish their various characteristic properties. Finally, we study the relationships between all these mappings and other weaker forms of fuzzy continuous mapping and introduce fuzzy sp-connected. Counter examples are given to show the noncoincidence of these sets and mappings.


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## Body

## 1. Introduction

In 1996, Dontchev and Przemski, [1] have introduced the concept of sp-open sets in general topology. In this paper, we extend the notion of sp-open sets to fuzzy topology space and study some notions based on this new concept. We further study the relation between fuzzy sp-open sets and other types of fuzzy open sets. We also introduce the concepts of fuzzy sp-continuous (resp., fuzzy sp-open) mapping, other stronger forms of fuzzy sp-continuous (resp., fuzzy sp-open) mapping, and discuss their relation with other weaker forms of fuzzy continuous mapping.


## 2. Preliminaries

Throughout this paper, by(X,τ) or simply by X we mean a fuzzy topological space (fts, shorty) and f:X→Y means a mapping f from a fuzzy topological space X to a fuzzy topological space Y. If u is a fuzzy set and p is a fuzzy singleton in X, then N(p), Intλ, clu, uc denote, respectively, the neighborhood system of p, the interior of u, the closure of u, and complement of u.Now, we recall some of the basic definitions and results in fuzzy topology.Definition 2.1 (see [2]).
A fuzzy singletonp in X is a fuzzy set defined by: p(x)=t, for x=x0 and p(x)=0 otherwise, where 0<t≤1. The point p is said to have support x0 and value t.Definition 2.2.
A fuzzy setu in a fts X is called fuzzy α-open [3] (resp., Fuzzy preopen [4], Fuzzy β-open [5]) set if u≤ Int cl Int u (resp., u≤ Int cl u,  u≤ cl Int cl u,  u≤ cl Int u). The family of all fuzzy α-open (resp., fuzzy preopen, fuzzy β-open, fuzzy semiopen) sets of X is denoted by FαO(X) (resp., FPO(X), FβO(X), FSO(X)).Definition 2.3 (see [4]).
Letu be any fuzzy set. Then,(i)
pclu=∧{v:v≥u,vis a fuzzy preclosed set ofX}is called preclosure,(ii)
pIntv=∨{v:v≤u,vis a fuzzy preopen set ofX} is called pre-Interior.The definitions of scl, sInt, α-cl, and α-Int are similar.Theorem 2.4.
For any fuzzy setu in a ftsX, the following statements are true:(i)
sclu=u∨Int clu and sIntu=u∧cl Intu [6],(ii)
pclu≥u∨cl IntuandpIntu≤u∧Int clu [7].Theorem 2.5 (see [3, 4]).
The arbitrary union of fuzzy preopen (resp., fuzzy semiopen) sets is a fuzzy preopen (resp., fuzzy semiopen) set.Definition 2.6.
A mappingf:(X,τ)→(Y,σ) is said to be(i)
fuzzyα-continuous [3] if f-1(v) is fuzzy α-open set in X for each fuzzy open set v in Y,(ii)
fuzzy semi continuous [3] if f-1(v) is fuzzy semiopen set in X for each fuzzy open set v in Y,(iii)
fuzzyβ-continuous [5] if f-1(v) is fuzzy β-open set in X for each fuzzy open set v in Y.


## 3. FuzzySP-Open Set

Definition 3.1.
A fuzzy subsetu of fuzzy space X is called fuzzy sp-open set if u≤Int clu∨cl Intu. The class of all fuzzy sp-open sets in X will be denoted be FSP-O(X).
It is obvious thatFPO(X)∨FSO(X)≤FSP-O(X)≤FβO(X).Proposition 3.2.
Letu be fuzzy sp -open set such that Intu=0. Then, u is fuzzy preopen.Using Theorem2.5, we can easily prove the next corollary.Corollary 3.3.
Any union of fuzzy sp -open sets is a fuzzy sp-open set.Remark 3.4.
The intersection of fuzzysp-open sets need not be fuzzy sp-open set. This is illustrated by the following example.Example 3.5.
LetX={a,b,c} and v1,v2,v3, and v4 be fuzzy sets of X defined as
(1)v1(a)=0.0,v1(b)=0.0,v1(c)=0.4,v2(a)=0.9,v2(b)=0.6,v2(c)=0.0,v3(a)=0.0,v3(b)=0.3,v3(c)=0.4,v4(a)=0.9,v4(b)=0.7,v4(c)=0.2.
Let τ={ox,v1,v2,v1∨v2,1x}. Clearly, τ is a fuzzy topology on X, and by easy computation, it follows that v3 and v4 are fuzzy sp-open sets. But (v3∧v4) is not a fuzzy fuzzy sp-open set.Theorem 3.6.
For any fuzzy subsetu of a fuzzy space X, the following properties are equivalent:(i)
u is fuzzy sp-open,(ii)
u≥pIntu∨sIntu.Proof.
(i)⇒(ii). Let u be a fuzzy sp-open, that is, u≤Int clu∨cl  Intu. Then, we have
(2)pIntu∨sIntu≤(u∧Int clu)∨(u∧cl Intu)≤u∧(Int clu∨cl  Intu)≤u∧u=u.Definition 3.7.
A fuzzy subsetu of fuzzy space X is called fuzzy sp-closed set if  Int clu∧cl Intu≤u. The class of all fuzzy sp-closed sets in X will be denoted be FSP-C(X).Definition 3.8.
Letu any fuzzy set. Then,(i)
sp-clu=∧{v:v≥u,vis a fuzzysp-closed set ofX} is called fuzzy sp-closure,(ii)
sp-Intv=∨{v:v≤u,vis a fuzzysp-open set ofX} is called fuzzy sp-Interior.By using Definitions 3.1, 3.7, and 3.8, we can prove the following theorems.Theorem 3.9.
Letu and v be the fuzzy sets in ftsX. Then, the following statements hold(i)
sp-cl(u) is fuzzy sp-closed,(ii)
u⊆SP-C(X)⇔u=sp-cl(u),(iii)
u≤v⇒sp-cl(u)≤sp-cl(v),(iv)
sp-Intu is fuzzy sp-open,(v)
u⊆SP-O(X)⇔u=sp-Int(u),(vi)
u≤v⇒sp-Int(u)≤sp-Int(v),(vii)
Intu≤sp-Intu≤u≤sp-cl(u)≤clu.Theorem 3.10.
For a fuzzy subsetλ of a fuzzy space X, the following statements are holding:(i)
sp-clu≥u∨(Int clu∧cl  Intu),(ii)
sp-Intu≤u∧(Int clu∨cl Intu),(iii)
sp-clu≥sclu∧pclu,(iv)
sp-Intu≤sIntu∨pIntu.Theorem 3.11.
For any fuzzy subsetu of a fuzzy space X, the following statements are equivalent:(i)
u is fuzzy sp-closed,(ii)
uc is fuzzy sp-open,(iii)
u≥Int  clu∧cl  Intu, and(iv)
uc≤Int  clu∨cl  Intu.Theorem 3.12.
A fuzzy setu in a fuzzy topology space X is fuzzy sp -open if and only if for every fuzzy point p∈u, there exists a fuzzy sp -open set vp≤u such that p∈vp.Proof.
Ifu is a fuzzy sp-open set, then we may take vp=u for every p∈u.
Conversely, we haveu=∪p∈u{p}≤∪p∈uvp≤u and, hence, u=∪p∈uvp. This shows that u is a fuzzy sp-open set.From Definitions2.2 and 3.1, the above “Implication Figure 1” illustrates the relation of different classes of fuzzy open sets.Figure 1Remark 3.13.
The converse of these relations need not be true, in general as shown by the following examples.Example 3.14.
LetX={a,b,c} and v1,v2,v3, and v4 be fuzzy sets of X defined as
(3)v1(a)=0.5,v1(b)=0.3,v2(a)=0.5,v2(b)=0.6,v3(a)=0.6,v3(b)=0.3,v4(a)=0.6,v4(b)=0.7.
Let τ={ox,v1,1x}. Clearly, τ is a fuzzy topological space on X, and by easy computation, we can see:(i)
v2 is fuzzy sp-open set which is neither fuzzy α-open set nor fuzzy preopen,(ii)
v3 is fuzzy sp-open which is not semiopen,(iii)
v4 is fuzzy sp-open set which is not fuzzy open.Example 3.15.
LetX={a,b,c} and v1,v2,andv3 fuzzy sets of X defined as
(4)v1(a)=0.0,v1(b)=0.0,v1(c)=0.4,v2(a)=0.9,v2(b)=0.6,v2(c)=0.0,v3(a)=0.1,v3(b)=0.0,v3(c)=0.3.
Let τ={ox,v1,1x}. Clearly, τ is a fuzzy topological space on X, and by easy computation, it follows that v3 is fuzzy β-open set which is not fuzzy sp-open.


## 4. FuzzySP-Continuous Mapping

Definition 4.1.
A mappingf:X→Y is said to be(i)
fuzzysp-continuous if f-1(v) is fuzzy sp-open set in X for each fuzzy open set v in Y,(ii)
fuzzysp⋆-continuous if f-1(v) is fuzzy sp-open set in X for each fuzzy sp-open set v in Y,(iii)
fuzzysp⋆⋆-continuous if f-1(v) is fuzzy open set in X for each fuzzy sp-open set v in Y.Theorem 4.2.
For a mappingf:(X,τ)→(Y,σ), the following statements are equivalent: (i)
f is  fuzzy sp-continuous;(ii)
for every fuzzy singletonp in X and every open set v in Y such that f(p)⊆v, there exists a fuzzy sp -open set u⊆X such that p⊆u and u≤f-1(v);(iii)
for every fuzzy singletonp in X and every open set v in Y such that f(p)⊆v, there exists a fuzzy sp -open set u⊆X such that p⊆u and f(u)≤v;(iv)
the inverse image of each fuzzy closed set inY is fuzzy sp-closed;(v)
Int cl(f-1(v))∧cl Int(f-1(v))≤(f-1(clv)) for each v⊆Y;(vi)
f[Int cl(u)∧cl Int(u)]≤clf(u) for each u⊆X.Proof.
(i)⇒(ii) Let fuzzy singleton p be in X and every open set v in Y such that f(p)⊆v, there exists a fuzzy open set m be in Y such that f(p)≤m≤v. Since f is sp-continuous, u=f-1(m) is fuzzy sp-open and we have p≤f-1(f(p))≤f-1(m)≤f-1(v)  or  p≤u=f-1(m)≤f-1(v).
(ii)⇒(iii) Let fuzzy singleton p be in X and every fuzzy open set v be in Y such that f(p)⊆v, there exists a fuzzy sp-open u such that p≤u and u≤f-1(v). So, we have p≤u and f(u)≤f-1((f(v))≤v.
(iii)⇒(i) Let v be a fuzzy open set in Y and let us take p≤f-1(v). This shows that f(p)≤f(f-1(v))≤v. Since v is a fuzzy open set, then there exists a fuzzy sp-open set u such that p≤u and f(u)≤v. This shows that p≤u≤f-1(f(u))≤f-1(v). By Theorem 3.12, it follows that f-1(v) is fuzzy sp-open set in X and hence f is fuzzy sp-continuous.
(i)⇒(iv) Let v be a fuzzy closed in Y. This implies that IY-v is fuzzy open set. Hence, f-1(IY-v) is fuzzy sp-open set in X, that is, (IX-f-1(v)) is fuzzy sp-open set in X. Thus, f-1(v) is a fuzzy sp-closed set in X.
(iv)⇒(v) Let v⊆Y, then f-1(clv) is sp-closed in X, that is, Int cl(f-1(v))∧cl Int(f-1(v))≤Int cl(f-1(clv))∧cl Int(f-1(clv))≤f-1(clv).
(v)⇒(vi) Let u⊆X, put v=f(u) in (v), then Int cl(f-1(f(u)))∧cl Int(f-1(f(u)))≤f-1(cl(f(u))) so that Int cl(u))∧cl Int(u)≤f-1(cl(f(u))). This gives f[Int cl(u)∧cl Int(u)]≤clf(u).
(vi)⇒(i) Let v⊆Y, be fuzzy open set. put u=IY-v and u=f-1(v), then f[Int cl(f-1(v))∧cl Int(f-1(v))]≤clf(f-1(v))≤cl(v)=v, that is, f-1(v)sp-closed set in X, so f is sp-continuous mapping.Definition 4.3.
A mappingf:X→Y is said to be(i)
fuzzysp-open (Fuzzy sp-closed) if f(u) is fuzzy sp-open (fuzzy sp-closed) set in Y for each fuzzy open (fuzzy closed) set u in X,(ii)
fuzzysp⋆-open (Fuzzy sp⋆-closed) if f(u) is fuzzy sp-open (fuzzy sp-closed) set in Y for each fuzzy sp-open (fuzzy sp-closed) set u in X,(iii)
fuzzysp⋆⋆-open (Fuzzy sp⋆⋆-closed) if f(u) is fuzzy open (fuzzy closed) set in Y for each fuzzy sp-open (fuzzy sp-closed) set u in X.Remark 4.4.
Iff:X→Y is fuzzy sp-continuous mapping and g:Y→Z is fuzzy sp-continuous mapping, then gof:X→Z may not be a fuzzy sp-continuous mapping; this can be show by the following example.Example 4.5.
LetX={a,b,c} and v1,v2,v3,v4 and v5 be fuzzy sets of X defined as,
(5)v1(a)=0.4,v1(b)=0.6,v1(c)=0.5,v2(a)=0.6,v2(b)=0.4,v2(c)=0.4,v3(a)=0.4,v3(b)=0.6,v3(c)=0.6,v4(a)=0.7,v4(b)=0.7,v4(c)=0.8,v5(a)=0.6,v5(b)=0.4,v5(c)=0.6.
Consider, ftsτ1, τ2, and τ3 where τ1={ox,v1,v2,v1∩v2,v1∪v2,1x}, τ2={ox,v4,1x}, and τ3={ox,v3,1x} and the mapping f:(X,τ1)→(Y,τ2) and g:(X,τ2)→(Y,τ3) defined as f(a)=b,f(b)=a,f(c)=c and g(a)=a,g(b)=b,g(c)=c. It is clear that f and g are fuzzy sp-continuous mapping. But gof:(X,τ1)→(Y,τ3) is not a fuzzy sp-continuous mapping. This because (gof)-1(v3)=v5 and v5 is not fuzzy sp-open set, and hence gof is not fuzzy sp-continuous mapping.Theorem 4.6.
Iff:X→Y is fuzzy sp -continuous mapping and g:Y→Z is fuzzy continuous mapping, then gof:X→Z is fuzzy sp-continuous mapping.Proof.
Letv be a fuzzy set of Z. Then, (gof)-1(v)=f-1(g-1(v)). And because g is fuzzy continuous this implies that g-1(v) is a fuzzy open set of Y and hence f-1(g-1(v)) is a fuzzy sp-open set in X. Therefore, gof is a fuzzy sp- continuous mapping.From Definitions4.1 and 4.3, we can have the above “Implication Figure 2” illustrates the relation between different classes of fuzzy sp-continuous (fuzzy semi sp-open) mappings.Figure 2The above “Implication Figure3” illustrates the relation between fuzzy sp-continuous and different classes of fuzzy continuous mapping.Figure 3Remark 4.7.
We can see the converse of these relations need not be true, in general as shown by the following examples.Example 4.8.
LetX={a,b,c} and v1,v2,v3, and v4 fuzzy sets of X defined as
(6)v1(a)=0.4,v1(b)=0.6,v1(c)=0.5,v2(a)=0.4,v2(b)=0.4,v2(c)=0.4,v3(a)=0.6,v3(b)=0.4,v3(c)=0.6,v4(a)=0.7,v4(b)=0.7,v4(c)=0.8.
Consider ftsτ1, τ2, and τ3 where τ1={ox,v1,v2,v1∩v2,v1∪v2,1x}, τ2={ox,v4,1x}, and τ3={ox,v3,1x} and the mapping f:(X,τ1)→(Y,τ2) and g:(X,τ1)→(Y,τ3) defined as f(a)=a,f(b)=b,f(c)=c and g(a)=a,g(b)=b,g(c)=c. It is clear that:(i)
f is a fuzzy sp-continuous mapping which is neither fuzzy α-continuous mapping nor fuzzy semi continuous mapping,(ii)
g is a fuzzy β-continuous mapping which is not fuzzy sp-continuous mapping,(iii)
f is a fuzzy sp-continuous mapping which is neither fuzzy sp⋆-continuous mapping nor fuzzy sp⋆⋆-continuous mapping, and this is because v3 is fuzzy sp-open in τ2 which is neither fuzzy sp-open nor fuzzy open in τ1,(iv)
ifh:(X,τ3)→(Y,τ1)  defined as: h(a)=a,h(b)=b,h(c)=c,  it is clear that h is a fuzzy sp-open mapping which is neither fuzzy sp⋆-open mapping nor fuzzy sp⋆⋆-open mapping.Definition 4.9.
A fuzzy setu in an ftsX is said to be fuzzy connected if u cannot be expressed as the union of two fuzzy separated sets.Now, we can generalize the definition of fuzzy connected to define fuzzysp-connected as follows.Definition 4.10.
A fuzzy setv in a fts(X,τ) is said to be fuzzy sp-connected if and only if v cannot be expressed as the union of two fuzzy sp-separated sets.Theorem 4.11.
Letf:X→Y be a fuzzy sp -continuous surjective mapping. If v is a fuzzy sp -connected subset in X then, f(v) is fuzzy connected in Y.Proof.
Suppose thatf(m) is not connected in Y. Then, there exist fuzzy separated subsets u and v in Y such that f(m)=u∪v.
Sincef is fuzzy sp-continuous surjective mapping, f-1(u) and f-1(v) are fuzzy sp-open set in X and m=f-1(f(m))=f-1(u∪v)=f-1(u)∪f-1(v).
It is clear thatf-1(u) and f-1(v) are fuzzy sp-separated in X. Therefore, m is not fuzzy sp-connected in X, which is a contradiction!!
Hence,Y is fuzzy connected.



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*Source: 768028-2011-04-10.xml*