Abstract

Several properties of fuzzy semiclosure and fuzzy semi-interior of fuzzy sets defined by Yalvac (1988), have been established and supported by counterexamples. We also study the characterizations and properties of fuzzy semi-open and fuzzy semi-closed sets. Moreover, we define fuzzy s-open and fuzzy s-closed mappings and give some interesting characterizations.

1. Introduction

The concept of fuzzy set was introduced by Zadeh in his classical paper [1]. This concept provides a natural foundation for treating mathematically the fuzzy phenomena, which exist pervasively in our real world, and for building new branches of fuzzy mathematics. In the area of Fuzzy Topology, introduced by Chang [2], much attention has been paid to generalize the basic concepts of General Topology in fuzzy setting and thus a modern theory of Fuzzy Topology has been developed.

In recent years, Fuzzy Topology has been found to be very useful in solving many practical problems. Du et al. [3] fuzzified the very successful 9-intersection Egenhofer model [4, 5] for depicting topological relations in Geographic Information Systems (GIS) query. In [6, 7], El Naschie showed that the notion of Fuzzy Topology might be relevant to quantum particle physics and quantum gravity in connection with string theory and š‘’āˆž theory. Tang [8] used a slightly changed version of Changā€™s fuzzy topological space to model spatial objects for GIS databases and Structured Query Language (SQL) for GIS.

Levine [9] introduced the concepts of semi-open sets and semicontinuous mappings in topological spaces. Interestingly, his work found applications in the field of Digital Topology [10]. For example, it was found that the digital line is a š‘‡1/2-space [11], which is a weaker separation axiom based upon semi-open sets. Fuzzy Digital Topology [12] was introduced by A. Rosenfeld, which demonstrated the need for the fuzzification of weaker forms of notions of Classical Topology. Azad [13] carried out this fuzzification in 1981, and presented some general properties of fuzzy spaces. Several properties of fuzzy semi-open (resp., fuzzy semi-closed), fuzzy regular open (resp., closed) sets have been discussed. Moreover he defined fuzzy semicontinuous (resp., semi-open and semi-closed) functions and studied the properties of fuzzy semicontinuous function in product related spaces. Finally, he defined and characterized fuzzy almost continuous mappings. For related subsequent work in this direction, we refer to [14ā€“27].

In this paper, our aim is to further contribute to the study of fuzzy semi-open and fuzzy semi-closed sets defined by Yalvac [26] by establishing several important fundamental identities and inequalities supported by counterexamples. Cameron and Woods [28] introduced the concepts of s-continuous mappings and s-open mappings. They investigated the properties of these mappings and their relationships to properties of semi-open sets. Khan and Ahmad [29] further worked on the characterizations and properties of s-continuous, s-open and s-closed mappings. We fuzzify the findings of [28, 29]. We define fuzzy s-open and fuzzy s-closed mappings and establish some interesting characterizations of these mappings.

2. Preliminaries

In order to make this paper self-contained, we briefly recall certain definitions and results; for those not described, we refer to [1, 2, 13, 26].

Let š‘‹={š‘„} be a space of points (objects), with a generic element š‘„. A fuzzy set šœ† in š‘‹ is characterized by membership function šœ†(š‘„) from š‘‹ to the unit interval [0,1].

The symbol Ī¦ denotes the empty fuzzy set defined as šœ‡Ī¦(š‘„)=0, for all š‘„āˆˆš‘‹. For š‘‹, the membership function is defined as šœ‡š‘‹(š‘„)=1, for all š‘„āˆˆš‘‹.

Definition 1 (see [2]). Let š‘“āˆ¶š‘‹ā†’š‘Œ be a mapping. Let š›½ be a fuzzy set in š‘Œ with membership function š›½(š‘¦). Then the inverse of š›½, written as š‘“āˆ’1(š›½), is a fuzzy set in š‘‹ whose membership function is defined by š‘“āˆ’1(š›½)(š‘„)=š›½(š‘“(š‘„)),āˆ€š‘„āˆˆš‘‹.(1) Conversely, let šœ† be a fuzzy set in š‘‹ with membership function šœ†(š‘„). The image of šœ†, written as š‘“(šœ†), is a fuzzy set in š‘Œ whose membership function is given by š‘“īƒÆ(šœ†)(š‘¦)=supš‘„āˆˆš‘“āˆ’1(š‘¦){šœ†(š‘„)},ifš‘“āˆ’1(š‘¦)isnonempty,0,otherwise,(2) for all š‘¦āˆˆš‘Œ.

Definition 2 (see [2]). A fuzzy topology is a family šœ of fuzzy sets in š‘‹, which satisfies the following conditions:(1)Ī¦,š‘‹āˆˆšœ;(2)If šœ†,šœ‡āˆˆšœ, then šœ†ā‹€šœ‡āˆˆšœ;(3)If šœ†š‘–āˆˆšœ for each š‘–āˆˆš¼, then āˆØš‘–šœ†š‘–āˆˆšœ.

šœ is called a fuzzy topology for š‘‹, and the pair (š‘‹,šœ) is an fts. Every member of šœ is called šœ-open fuzzy set (or simply an open fuzzy set). A fuzzy set is šœ-closed if and only if its complement is šœ-open.

As in general topology, the indiscrete fuzzy topology contains only Ī¦ and š‘‹, while the discrete fuzzy topology contains all fuzzy sets.

3. Fuzzy Semi-Open and Fuzzy Semi-Closed Sets

ā€‰

Definition 3 (see [13]). Let šœ† be a fuzzy set in an fts (š‘‹,šœ). Then šœ† is called a fuzzy semi-open set of š‘‹, if there exists a šœˆāˆˆšœ such that šœˆā‰¤šœ†ā‰¤Clšœˆ. A fuzzy set šœ‡ is fuzzy semi-closed if and only if its complement šœ‡š‘ is fuzzy semi-open. The class of all fuzzy semi-open (resp., fuzzy semi-closed) sets in š‘‹ is denoted by FSO(š‘‹) (resp., FSC(š‘‹)).

Definition 4 (see [26]). Let šœ† be a fuzzy set in an fts š‘‹. Then semi-closure (briefly sCl) and semi-interior (briefly sInt) of šœ† are defined as sClšœ†=āˆ§{š›½āˆ£šœ†ā‰¤š›½,š›½fuzzysemi-closed},sIntšœ†=āˆØ{š›½āˆ£š›½ā‰¤šœ†,š›½fuzzysemi-open},(3) and are called the fuzzy semi-closure of šœ† and fuzzy semi-interior of šœ†, respectively.

It is immediate that

(1)sClšœ†ā‰„šœ† and sIntšœ†ā‰¤šœ†;(2)šœ†ā‰¤šœ‡ā‡’sClšœ†ā‰¤sClšœ‡, sIntšœ†ā‰¤sIntšœ‡.

It is known [13] that a fuzzy set šœ† in an fts š‘‹ is

(1)fuzzy semi-closed if and only if IntClšœ†ā‰¤šœ† (resp., sClšœ†=šœ†);(2)fuzzy semi-open if and only if šœ†ā‰¤ClIntšœ† (resp., sIntšœ†=šœ†).

The following are characterizations of fuzzy semi-closed sets, the proof of Theorem 1 is straightforward.

Theorem 1. A fuzzy set šœ† is fuzzy semi-closed if and only if there exists a fuzzy closed set šœ“ such that Intšœ“ā‰¤šœ†ā‰¤šœ“.

Theorem 2. For a fuzzy set šœ† in an fts š‘‹, šœ† is fuzzy semi-closed if and only if sIntsClšœ†ā‰¤šœ† .

Proof. (ā‡’) If šœ†isafuzzysemi-closedset,thatis,šœ†=sClšœ†, then sIntsClšœ†=sIntšœ†ā‰¤šœ†.
(ā‡) Suppose sInt(sClšœ†)ā‰¤šœ†. Since sClšœ† is fuzzy semi-closed, so there exists a closed set šœ“ such that Intšœ“ā‰¤sClšœ†ā‰¤šœ“. Thus Intšœ“ā‰¤sIntsClšœ†ā‰¤šœ†ā‰¤sClšœ†ā‰¤šœ“ or Intšœ“ā‰¤šœ†ā‰¤šœ“. Hence šœ† is fuzzy semi-closed and šœ†=sClšœ†.

Remark 1. It is easily seen that(1)if šœ† is fuzzy semi-open (resp., fuzzy semi-closed), then Intšœ†,sIntšœ†,sClšœ†, and Clšœ† are fuzzy semi-open (resp., fuzzy semi-closed);(2)a nonvoid nowhere dense fuzzy set šœ† is fuzzy semi-closed and not fuzzy semi-open.

The converse of Remark 1(2) is, in general, not true as is shown by following.

Example 1. Let š‘‹={š‘Ž,š‘,š‘} be a set and š¼={0,.3,.5,.7,1} be the lattice of membership grades for fuzzy sets in š‘‹. Let šœ‡={š‘Ž.7,š‘0,š‘1},šœˆ={š‘Ž.7,š‘.5,š‘.3}, and šœ”={š‘Ž.5,š‘.5,š‘.5} be fuzzy sets on š‘‹, and šœ the fuzzy topology generated by šœ‡,šœˆ, and šœ”. Then šœ={Ī¦,šœ‡,šœˆ,šœ”,{š‘Ž.5,š‘0,š‘.5},{š‘Ž.5,š‘.5,š‘.3}, {š‘Ž.7,š‘0,š‘.3},{š‘Ž.7,š‘0,š‘.5},{š‘Ž.7,š‘.5,š‘.5},{š‘Ž.7,š‘.5,š‘1},š‘‹}. Calculations give that fuzzy set šœ” is both fuzzy semi-closed and fuzzy semi-open but IntClšœ”=šœ”.

Theorem 3. For fuzzy sets šœ† and šœ‡ in an fts š‘‹, one has (1)š‘ š¼š‘›š‘”(šœ†āˆØšœ‡)ā‰„š‘ š¼š‘›š‘”šœ†āˆØš‘ š¼š‘›š‘”šœ‡;(2)ā‹€ā‹€š‘ š¼š‘›š‘”(šœ†šœ‡)=š‘ š¼š‘›š‘”šœ†š‘ š¼š‘›š‘”šœ‡;(3)š‘ š¶š‘™(šœ†āˆØšœ‡)=š‘ š¶š‘™šœ†āˆØš‘ š¶š‘™šœ‡;(4)ā‹€ā‹€š‘ š¶š‘™(šœ†šœ‡)ā‰¤š‘ š¶š‘™šœ†š‘ š¶š‘™šœ‡.

Proof. (1) sIntšœ† and sIntšœ‡ are both fuzzy semi-open . šœ†ā‰¤šœ†āˆØšœ‡,šœ‡ā‰¤šœ†āˆØšœ‡ imply sIntšœ†ā‰¤sInt(šœ†āˆØšœ‡) and sIntšœ‡ā‰¤sInt(šœ†āˆØšœ‡). Combining, sIntšœ†āˆØsIntšœ‡ā‰¤sInt(šœ†āˆØšœ‡) or ī˜sInt(šœ†āˆØšœ‡)ā‰„sIntšœ†sIntšœ‡.(4)
(2) šœ†ā‹€šœ‡ā‰¤šœ† and šœ†ā‹€šœ‡ā‰¤šœ‡ imply ā‹€ā‹€sInt(šœ†šœ‡)ā‰¤sIntšœ†sIntšœ‡. Conversely sIntšœ†ā‰¤šœ† and sIntšœ‡ā‰¤šœ‡ imply ā‹€ā‹€šœ‡sIntšœ†sIntšœ‡ā‰¤šœ† and ā‹€sIntšœ†sIntšœ‡ is fuzzy semi-open. But ā‹€sInt(šœ†šœ‡) is the largest fuzzy semi-open set such that ā‹€ā‹€sInt(šœ†šœ‡)ā‰¤šœ†šœ‡, hence ā‹€ā‹€sIntšœ†sIntšœ‡ā‰¤sInt(šœ†šœ‡). This gives the equality.
(3) and (4) follow easily from (2).

The inequalities (1) and (4) of Theorem 3, are in general irreversible, as is shown by following.

Example 2. Let š‘‹={š‘Ž,š‘} be a set and š¼={0,.5,1} be the lattice of membership grades for fuzzy sets on š‘‹. Let šœ‡={š‘Ž.5,š‘1},šœˆ={š‘Ž.0,š‘1}, and šœ”={š‘Ž1,š‘.5} be fuzzy sets on š‘‹, and šœ the fuzzy topology generated by šœ‡, šœˆ, and šœ”. Then ī€½ī€½š‘Žšœ=Ī¦,šœ‡,šœˆ,šœ”,.5,š‘.5ī€¾,ī€½š‘Ž0,š‘.5ī€¾ī€¾,š‘‹.(5) We choose fuzzy sets š›¼={š‘Ž.5,š‘0},š›½={š‘Ž0,š‘.5},š›¾={š‘Ž.5,š‘1} and š›æ={š‘Ž1,š‘.5}. Then calculations give that ī€½š‘ŽsInt(š›¼āˆØš›½)=.5,š‘.5ī€¾Ģø<ī€½š‘Ž0,š‘.5ī€¾ī€½š‘Ž=sIntš›¼āˆØsIntš›½,sClš›¾āˆ§sClš›æ=1,š‘.5ī€¾Ģø<ī€½š‘Ž.5,š‘.5ī€¾=sCl(š›¾āˆ§š›æ).(6)

It is known [30] that for a fuzzy set šœ† in an fts š‘‹, we have

(1)ClsClšœ†=Clšœ†;(2)IntClšœ†ā‰¤sClšœ†.

We use this and Theorem 1, and prove the following.

Theorem 4. Let šœ† be a fuzzy set in an fts š‘‹. Then one has the following: (1)š‘ š¶š‘™š‘ š¶š‘™šœ†=š‘ š¶š‘™šœ†(š‘ š¼š‘›š‘”š‘ š¼š‘›š‘”šœ†=š‘ š¼š‘›š‘”šœ†);(2)š‘ š¼š‘›š‘”š‘ š¶š‘™šœ†ā‰„š¼š‘›š‘”š¶š‘™šœ†;(3)š¼š‘›š‘”š‘ š¼š‘›š‘”š‘ š¶š‘™šœ†=š¼š‘›š‘”š¶š‘™šœ†;(4)š‘ š¼š‘›š‘”š‘ š¶š‘™šœ†ā‰¤š¼š‘›š‘”š¶š‘™šœ†āˆØšœ†.

Proof. (1) By the fact that sClšœ† is fuzzy semi-closed and that šœ† is fuzzy semi-closed if and only if sClšœ†=šœ†, it follows immediately.
(2) Since , IntClšœ†ā‰¤sClšœ†, so that IntClšœ†ā‰¤sIntsClšœ†, since all fuzzy open sets are fuzzy semi-open.
(3) By (2), IntClšœ†ā‰¤sIntsClšœ†, so that IntClšœ†ā‰¤IntsIntsClšœ†. Also sClšœ†ā‰¤Clšœ†, so that sIntsClšœ†ā‰¤Clšœ† and IntsIntsClšœ†ā‰¤IntClšœ†. Consequently, we get (3).
(4) Now IntClšœ†ā‰¤IntClšœ†āˆØšœ†ā‰¤Clšœ† so by Theorem 1, IntClšœ†āˆØšœ† is fuzzy semi-closed. Also šœ†ā‰¤IntClšœ†āˆØšœ†. Since sClšœ† is the smallest fuzzy semi-closed set with šœ†ā‰¤sClšœ†, therefore, sClšœ†ā‰¤IntClšœ†āˆØšœ†. Which implies sIntsClšœ†ā‰¤sClšœ†ā‰¤IntClšœ†āˆØšœ† or sIntsClšœ†ā‰¤IntClšœ†āˆØšœ†.

The inequalites (2) and (4) of Theorem 4 may, in general, not be reversible as is shown in the following.

Example 3. Let š‘‹={š‘Ž,š‘,š‘} be a set and š¼={0,.3,.5,.7,1} be the lattice of membership grades for fuzzy sets on š‘‹. Let šœ‡={š‘Ž.5,š‘1,š‘.3}, šœˆ={š‘Ž0,š‘.5,š‘1}, and šœ”={š‘Ž.3,š‘.7,š‘.5} be fuzzy sets on š‘‹, and šœ the fuzzy topology generated by šœ‡, šœˆ, and šœ”. Then šœ={Ī¦,šœ‡,šœˆ,šœ”,{š‘Ž0,š‘.5,š‘.5},{š‘Ž0,š‘.5,š‘.3}, {š‘Ž.3,š‘.7,š‘.3},{š‘Ž.5,š‘1,š‘.5},{š‘Ž.3,š‘.7,š‘1},{š‘Ž.5,š‘1,š‘1},š‘‹}.
We choose fuzzy sets šœ†={š‘Ž.7,š‘.5,š‘.5} and šœ“={š‘Ž0,š‘0,š‘.5}, then calculations give that ī€½š‘ŽsIntsClšœ†=.7,š‘.5,š‘.5ī€¾Ģø<ī€½š‘Ž0,š‘.5,š‘.5ī€¾ī€½š‘Ž=IntClšœ†;IntClšœ“āˆØšœ“=0,š‘0,š‘.5ī€¾Ģø<ī€½š‘Ž0,š‘0,š‘0ī€¾=sIntsClšœ“.(7)

Theorem 5. For any fuzzy set šœ† in an fts š‘‹, we have (1)(š‘ š¼š‘›š‘”šœ†)š‘=š‘ š¶š‘™(šœ†š‘),(2)(š‘ š¶š‘™šœ†)š‘=š‘ š¼š‘›š‘”(šœ†š‘).

Proof. (1) (sIntšœ†)š‘=1āˆ’sIntšœ†=1āˆ’āˆØ{šœ‡āˆ£šœ‡āˆˆFSO(š‘‹)andšœ‡ā‰¤šœ†}=āˆ§{1āˆ’šœ‡āˆ£šœ‡āˆˆFSO(š‘‹)andšœ‡ā‰¤šœ†}=āˆ§{šœ“āˆ£šœ“š‘āˆˆFSO(š‘‹)andšœ“ā‰„šœ†š‘},wherešœ“=1āˆ’šœ‡=sCl(šœ†š‘).(8)
(2) Similar to (1).

4. Fuzzy S-Open and Fuzzy S-Closed Mappings

First, we define

Definition 5. A function š‘“āˆ¶š‘‹ā†’š‘Œ is said to be fuzzy s-open (resp., fuzzy s-closed) if the image of every fuzzy semi-open (resp., fuzzy semi-closed) set is fuzzy open (resp., fuzzy closed).

Obviously a fuzzy s-open function is fuzzy open.

Definition 6 (see [31]). A fuzzy point š‘’ is called a boundary point of a fuzzy set šœ† if and only if ā‹€š‘’āˆˆClšœ†Clšœ†š‘. The union of all the boundary points of šœ† is called a boundary of šœ†, denoted by Bdšœ†. It is clear that Bdšœ†=Clšœ†āˆ§Clšœ†š‘.(9)

Next, we define

Definition 7. Semiboundary (briefly sBd) of a fuzzy set šœ† in an fts š‘‹ is defined as sBdšœ†=sClšœ†āˆ§sClšœ†š‘.(10)

In the following, we characterize fuzzy s-open mappings in terms of sInt, sCl, and sBd.

Theorem 6. For a function š‘“āˆ¶š‘‹ā†’š‘Œ, a fuzzy set š›¼ in an fts š‘‹ and a fuzzy set š›½ in an fts š‘Œ, then the following are equivalent: (1)š‘“ is fuzzy s-open;(2)š‘“(sIntš›¼)ā‰¤Intš‘“(š›¼);(3)sIntš‘“āˆ’1(š›½)ā‰¤š‘“āˆ’1(Intš›½);(4)š‘“āˆ’1(Clš›½)ā‰¤sCl(š‘“āˆ’1(š›½));(5)š‘“āˆ’1(Bd(š›½))ā‰¤sBd(š‘“āˆ’1(š›½)).

Proof. (1)ā‡’(2) obviously š‘“(sIntš›¼)ā‰¤š‘“(š›¼). š‘“ is fuzzy s-open gives š‘“(sIntš›¼)that is fuzzy open in š‘Œ. But Intš‘“(š›¼) is the largest fuzzy open set such that Intš‘“(š›¼)ā‰¤š‘“(š›¼). Therefore š‘“(sIntš›¼)ā‰¤Intš‘“(š›¼), for any fuzzy set š›¼ in š‘‹. This gives (2).
(2)ā‡’(3) For any fuzzy set š›½ in š‘Œ, š‘“āˆ’1(š›½)=š›¼ is a fuzzy set in š‘‹. Then by (2)š‘“ī€·sIntš‘“āˆ’1ī€øī€·š‘“ī€·š‘“(š›½)ā‰¤Intāˆ’1ī€ø(š›½)ā‰¤Int(š›½),(11) or š‘“(sIntš‘“āˆ’1(š›½))ā‰¤Intš›½ or sIntš‘“āˆ’1(š›½)ā‰¤š‘“āˆ’1š‘“(sIntš‘“āˆ’1(š›½))ā‰¤š‘“āˆ’1(Intš›½) or sIntš‘“āˆ’1(š›½)ā‰¤š‘“āˆ’1(Intš›½). This gives (3).
(3)ā‡’(4) By (3), we have sIntš‘“āˆ’1(š›½)ā‰¤š‘“āˆ’1(Intš›½)(š‘“āˆ’1(Intš›½))š‘ā‰¤(sIntš‘“āˆ’1(š›½))š‘ī€·š‘“=sClāˆ’1ī€ø(š›½)š‘(byTheorem5(1)),(12) or (š‘“āˆ’1(Intš›½))š‘ā‰¤sCl(š‘“āˆ’1(š›½))š‘ or š‘“āˆ’1(Clš›½š‘)ā‰¤sClš‘“āˆ’1(š›½š‘) or š‘“āˆ’1(Clšœ“)ā‰¤sClš‘“āˆ’1(šœ“) where šœ“=š›½š‘, a fuzzy set in š‘Œ. This gives (4).
(4)ā‡’(5) For a fuzzy set š›½ in š‘Œ, ā‹€Bdš›½=Clš›½Cl(š›½š‘) is a fuzzy closed set in š‘Œ. Now š‘“āˆ’1(Bdš›½)=š‘“āˆ’1ā‹€š‘“(Clš›½)āˆ’1(Clš›½š‘). Using (4) we have š‘“āˆ’1ī€·š‘“(Bdš›½)ā‰¤sClāˆ’1ī€·š‘“(š›½)āˆ§sClāˆ’1(š›½š‘)ī€ø(13) or š‘“āˆ’1(Bdš›½)ā‰¤sClš‘“āˆ’1ī€·š‘“(š›½)āˆ§sClāˆ’1ī€ø(š›½)š‘=sBdš‘“āˆ’1(š›½).(14) This gives (5).

In the following, we give characterizations of fuzzy s-closed mappings as follows.

Theorem 7. A function š‘“āˆ¶š‘‹ā†’š‘Œ is fuzzy s-closed if and only if Clš‘“(šœ†)ā‰¤š‘“(sClšœ†), for each fuzzy set šœ† in an fts š‘‹.

Proof. (ā‡’) Obviously š‘“(šœ†)ā‰¤š‘“(sClšœ†). š‘“(sClšœ†) is fuzzy closed, since š‘“ is fuzzy s-closed. But Clš‘“(šœ†) is the smallest fuzzy closed set with š‘“(šœ†)ā‰¤Clš‘“(šœ†). Therefore Clš‘“(šœ†)ā‰¤š‘“(sClšœ†).
(ā‡) Let šœ†āˆˆFSC(š‘‹). We show that š‘“(šœ†) is fuzzy closed. By hypothesis, Clš‘“(šœ†)ā‰¤š‘“(sClšœ†)=š‘“(šœ†) or Clš‘“(šœ†)ā‰¤š‘“(šœ†). This proves that š‘“(šœ†) is fuzzy closed.

Theorem 8. If a function š‘“āˆ¶š‘‹ā†’š‘Œ is fuzzy s-closed then for each fuzzy set š›½ in an fts š‘Œ and each fuzzy semi-open set šœ‡ in an fts š‘‹ with šœ‡ā‰„š‘“āˆ’1(š›½), there exists a fuzzy open set šœˆ in š‘Œ with šœˆā‰„š›½ such that š‘“āˆ’1(šœˆ)ā‰¤šœ‡.

Proof. Let šœ‡ be an arbitrary fuzzy semi-open set in š‘‹ with šœ‡ā‰„š‘“āˆ’1(š›½), where š›½ is a fuzzy set in š‘Œ. Clearly (š‘“(šœ‡š‘))š‘=šœˆ (say) is fuzzy open in š‘Œ. Since š‘“āˆ’1(š›½)ā‰¤šœ‡, then straightforward calculations give that š›½ā‰¤šœˆ. Moreover, we have š‘“āˆ’1(šœˆ)=š‘“āˆ’1(š‘“(šœ‡š‘))š‘=ī€·š‘“āˆ’1š‘“(šœ‡š‘)ī€øš‘ā‰¤šœ‡orš‘“āˆ’1(šœˆ)ā‰¤šœ‡.(15)

Theorem 9. Let š‘“āˆ¶š‘‹ā†’š‘Œ be a surjective function from an fts š‘‹ to an fts š‘Œ. If for each fuzzy set š›½ in š‘Œ and each fuzzy semi-open set šœ‡ in š‘‹ with šœ‡ā‰„š‘“āˆ’1(š›½), there exists a fuzzy open set šœˆ in š‘Œ with šœˆā‰„š›½ such that š‘“āˆ’1(šœˆ)ā‰¤šœ‡, then š‘“ is s-closed.

Proof. Let šœ“ be an arbitrary fuzzy semi-closed set in š‘‹ and š‘¦āˆˆ(š‘“(šœ“))š‘. Then š‘“āˆ’1(š‘¦)ā‰¤š‘“āˆ’1(š‘“(šœ“))š‘=ī€·š‘“āˆ’1ī€øš‘“(šœ“)š‘ā‰¤šœ“š‘(16) or š‘“āˆ’1(š‘¦)ā‰¤šœ“š‘. Since šœ“š‘ is fuzzy semi-open, therefore there exists a fuzzy open set šœˆš‘¦ with š‘¦āˆˆšœˆš‘¦ such that š‘“āˆ’1(šœˆš‘¦)ā‰¤šœ“š‘. Since š‘“ is surjective, we have š‘¦āˆˆšœˆš‘¦ā‰¤(š‘“(šœ“))š‘. Thus (š‘“(šœ“))š‘=āˆØ{šœˆš‘¦āˆ£š‘¦āˆˆ(š‘“(šœ“))š‘} is fuzzy open in š‘Œ or š‘“(šœ“) is fuzzy closed in š‘Œ. This proves that š‘“ is s-closed.

Combining Theorems 8 and 9, we have the following.

Theorem 10. A surjective function š‘“āˆ¶š‘‹ā†’š‘Œ is fuzzy s-closed if and only if for each fuzzy set š›½ in š‘Œ and each fuzzy semi-open set šœ‡ in š‘‹ with šœ‡ā‰„š‘“āˆ’1(š›½), there exists a fuzzy open set šœˆ in š‘Œ with šœˆā‰„š›½ such that š‘“āˆ’1(šœˆ)ā‰¤šœ‡.