Fuzzy Boundary and Fuzzy Semiboundary

Abstract

We present several properties of fuzzy boundary and fuzzy semiboundary which have been supported by examples. Properties of fuzzy semi-interior, fuzzy semiclosure, fuzzy boundary, and fuzzy semiboundary have been obtained in product-related spaces. We give necessary conditions for fuzzy continuous (resp., fuzzy semicontinuous, fuzzy irresolute) functions. Moreover, fuzzy continuous (resp., fuzzy semicontinuous, fuzzy irresolute) functions have been characterized via fuzzy-derived (resp., fuzzy-semiderived) sets.

1. Introduction

After Zadeh's [1] introduction of fuzzy sets, Chang [2] defined and studied the notion of a fuzzy topological space in 1968. Since then, much attention has been paid to generalize the basic concepts of classical 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 𝑒^∞ theory. Tang [8] used a slightly changed version of Chang's fuzzy topological space to model spatial objects for GIS databases and Structured Query Language (SQL) for GIS.

Levine [9] introduced the concepts of semiopen 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 digital line is a 𝑇_1/2-space [11], which is a weaker separation axiom based upon semiopen sets. Fuzzy digital topology [12] was introduced by 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 semiopen (resp., fuzzy semiclosed), fuzzy regular open (resp., closed) sets have been discussed. Moreover, he defined fuzzy semicontinuous (resp., semiopen, semiclosed) functions and studied the properties of fuzzy semicontinuous function in product-related spaces. Finally, he defined and characterized fuzzy almost continuous mappings. In this direction much work followed subsequently, for example, [14–23].

Though Pu and Liu [24] defined the notion of fuzzy boundary in fuzzy topological spaces in 1980, yet there is very little work available on this notion in present literature. One reason, inter alia, of Tang's [8] use of a limited version of Chang's fuzzy topological space was the nonavailability of sufficient material about properties of fuzzy boundary. So, we study this concept further and establish several of its properties, thus providing sufficient material for researchers to utilize these concepts fruitfully. Ahmad and Athar [25] defined the concept of fuzzy semiboundary and characterized fuzzy semicontinuous functions in terms of fuzzy semiboundary.

In this paper, we present several properties of fuzzy boundary and fuzzy semiboundary which have been supported by examples. Properties of fuzzy semi-interior, fuzzy semiclosure, fuzzy boundary, and fuzzy semiboundary have been obtained in product-related spaces. We give necessary conditions for fuzzy continuous (resp., fuzzy semicontinuous, fuzzy irresolute) functions. Moreover, fuzzy continuous (resp., fuzzy semicontinuous, fuzzy irresolute) functions have been characterized via fuzzy-derived (resp., fuzzy semiderived) sets.

2. Preliminaries

First, we briefly recall certain definitions and results; for those not described; we refer to [1, 2, 13, 22].

A fuzzy set 𝜆 in a set 𝑋 is a function from 𝑋 to [0,1], that is, 𝜆∶𝑋→[0,1].

Definition 1 (see [1]). Let 𝜆 and 𝜇 be fuzzy sets in 𝑋. Then, for all 𝑥∈𝑋, ,,𝜆=𝜇⟺𝜆(𝑥)=𝜇(𝑥),𝜆≤𝜇⟺𝜆(𝑥)≤𝜇(𝑥),𝜓=𝜆∨𝜇⟺𝜓(𝑥)=max𝜆(𝑥),𝜇(𝑥)𝛿=𝜆∧𝜇⟺𝛿(𝑥)=min𝜆(𝑥),𝜇(𝑥)𝜂=𝜆^𝑐⟺𝜂(𝑥)=1−𝜆(𝑥). (1)

More generally, for a family Λ={𝜆_𝑖∣𝑖∈𝐼} of fuzzy sets in 𝑋, the union 𝜓=∨_𝑖𝜆_𝑖 and intersection 𝛿=∧_𝑖𝜆_𝑖 are defined by𝜓(𝑥)=sup_𝑖𝜆_𝑖(𝑥)∣𝑥∈𝑋,𝛿(𝑥)=inf_𝑖𝛿_𝑖(𝑥)∣𝑥∈𝑋. (2)

The empty fuzzy set 0 is defined as 0(𝑥)=0, for all 𝑥∈𝑋, and the symbol 1 denotes the fuzzy set 1(𝑥)=1, for all 𝑥∈𝑋.

Definition 2 (see [2]). Let 𝑓∶𝑋→𝑌 be a function. Let 𝛽 be a fuzzy set in 𝑌 with membership function 𝛽(𝑦). Then, the inverse of 𝛽, written as 𝑓⁻¹(𝛽), is a fuzzy set in 𝑋 whose membership function is defined by𝑓⁻¹(𝛽)(𝑥)=𝛽𝑓(𝑥),∀𝑥∈𝑋. (3)Conversely, let 𝜆 be a fuzzy set in 𝑋 with membership function 𝜆(𝑥). The image of 𝜆, written as 𝑓(𝜆), is a fuzzy set in 𝑌 whose membership function is given by𝑓(𝜆)(𝑦)=sup_𝑥∈𝑓⁻¹_(𝑦)𝜆(𝑥),if𝑓⁻¹(𝑦)isnonempty,0,otherwise, (4)for all 𝑦∈𝑌, where 𝑓⁻¹(𝑦)={𝑥∣𝑓(𝑥)=𝑦}.

Definition 3 (see [2]). A fuzzy topology is a family 𝜏 of fuzzy sets in 𝑋, which satisfies the following conditions: (1)0,1∈𝜏;(2)if 𝜆,𝜇∈𝜏, then 𝜆∧𝜇∈𝜏;(3)if 𝜆_𝑖∈𝜏 for each 𝑖∈𝐼, then ∨_𝑖𝜆_𝑖∈𝜏.

𝜏 is called a fuzzy topology for 𝑋, and the pair (𝑋,𝜏) is a fuzzy topological space. Every member of 𝜏 is called 𝜏-open fuzzy set (or simply fuzzy open set). A fuzzy set is 𝜏-closed if and only if its complement is 𝜏-open.

As in general topology, the indiscrete fuzzy topology contains only 0 and 1, while the discrete fuzzy topology contains all fuzzy sets. In the sequel, we write an fts 𝑋 (or (𝑋,𝜏)) in place of “a space 𝑋 with fuzzy topology 𝜏”.

Definition 4 (see [2]). The closure and interior of a fuzzy set 𝜆 in an fts (𝑋,𝜏) are denoted and defined asCl𝜆=inf𝜇∣𝜆≤𝜇,𝜇^𝑐,.∈𝜏Int𝜆=sup𝜔∣𝜔≤𝜆,𝜔∈𝜏 (5)

We mention below some properties of closure and interior of a fuzzy set which will be used in the sequel.

Lemma 1 (see [26]). For fuzzy sets 𝜆 and 𝜇 in an fts 𝑋, one has the following: (1)𝜆 is fuzzy closed (resp., fuzzy open) ⇔𝐶𝑙𝜆=𝜆 (resp., 𝐼𝑛𝑡𝜆=𝜆);(2)𝜆≤𝜇⇒𝐶𝑙𝜆≤𝐶𝑙𝜇 (𝐼𝑛𝑡𝜆≤𝐼𝑛𝑡𝜇);(3)𝐶𝑙𝐶𝑙𝜆=𝐶𝑙𝜆 (𝐼𝑛𝑡𝐼𝑛𝑡𝜆=𝐼𝑛𝑡𝜆);(4)𝐶𝑙𝜆∨𝐶𝑙𝜇=𝐶𝑙(𝜆∨𝜇);(5)𝐶𝑙𝜆∧𝐶𝑙𝜇≥𝐶𝑙(𝜆∧𝜇);(6)𝐼𝑛𝑡𝜆∨𝐼𝑛𝑡𝜇≤𝐼𝑛𝑡(𝜆∨𝜇);(7)𝐼𝑛𝑡𝜆∧𝐼𝑛𝑡𝜇=𝐼𝑛𝑡(𝜆∧𝜇);(8)𝐼𝑛𝑡𝜆^𝑐=(𝐶𝑙𝜆)^𝑐;(9)𝐶𝑙𝜆^𝑐=(𝐼𝑛𝑡𝜆)^𝑐.

Definition 5 (see [2]). A function 𝑓∶(𝑋,𝜏)→(𝑌,𝛿) is said to be fuzzy continuous if and only if the inverse of each 𝛿-fuzzy open set is 𝜏-fuzzy open.

Theorem 1 (see [19, 26]). Let 𝑓∶𝑋→𝑌 be a function from an fts (𝑋,𝜏) to another fts (𝑌,𝛿). Then, the following conditions are equivalent: (1)𝑓 is fuzzy continuous,(2)the inverse of each 𝛿-closed fuzzy set is 𝜏-closed,(3)for each fuzzy set 𝜆 in 𝑋, 𝑓(𝐶𝑙𝜆)≤𝐶𝑙𝑓(𝜆), (4)for each fuzzy set 𝜆 in 𝑌, 𝑓⁻¹(𝐼𝑛𝑡𝜈)≤𝐼𝑛𝑡𝑓⁻¹(𝜈).

3. Fuzzy Boundary

Definition 6 (see [24]). Let 𝜆 be a fuzzy set in an fts 𝑋. Then, the fuzzy boundary of 𝜆 is defined as Bd𝜆=Cl𝜆∧Cl𝜆^𝑐. Obviously, Bd𝜆 is a fuzzy closed set.

Remark 1. In classical topology, for an arbitrary set 𝐴 of a topological space 𝑋, we have 𝐴∪Bd𝐴=Cl𝐴, but in fuzzy topology we have 𝜆∨Bd𝜆≤Cl𝜆, for an arbitrary fuzzy set 𝜆 in 𝑋, the converse of which is not true as shown by Pu and Liu [24]. Moreover, we have the following proposition.

Proposition 1. For fuzzy sets 𝜆 and 𝜇 in an fts 𝑋, the following conditions hold.
(1)𝐵𝑑𝜆=𝐵𝑑𝜆^𝑐.(2)If 𝜆 is fuzzy closed, then 𝐵𝑑𝜆≤𝜆.(3)If 𝜆 is fuzzy open, then 𝐵𝑑𝜆≤𝜆^𝑐.(4)Let 𝜆≤𝜇 and 𝜇∈𝐹𝐶(𝑋) (resp., 𝜇∈𝐹𝑂(𝑋)) . Then, 𝐵𝑑𝜆≤𝜇 (resp., 𝐵𝑑𝜆≤𝜇^𝑐), where 𝐹𝐶(𝑋) (resp., 𝐹𝑂(𝑋)) denotes the class of fuzzy closed (resp., fuzzy open) sets in 𝑋.(5)(𝐵𝑑𝜆)^𝑐=𝐼𝑛𝑡𝜆∨𝐼𝑛𝑡𝜆^𝑐.

Proof. (1) Bd𝜆=Cl𝜆∧Cl𝜆^𝑐=Cl𝜆^𝑐∧Cl𝜆=Cl𝜆^𝑐∧Cl(𝜆^𝑐)^𝑐=Bd𝜆^𝑐.
(2) Bd𝜆=Cl𝜆∧Cl𝜆^𝑐≤Cl𝜆=𝜆, hence Bd𝜆≤𝜆.
(3) 𝜆 is fuzzy open implies 𝜆^𝑐 is fuzzy closed. By (2), Bd𝜆^𝑐≤𝜆^𝑐 and by (1), we get Bd𝜆≤𝜆^𝑐.
(4) Since 𝜆≤𝜇 implies Cl𝜆≤Cl𝜇, we have Bd𝜆=Cl𝜆∧Cl𝜆^𝑐≤Cl𝜇∧Cl𝜆^𝑐≤Cl𝜇=𝜇, since 𝜇∈FC(𝑋).
(5) (Bd𝜆)^𝑐=(Cl𝜆∧Cl𝜆^𝑐)^𝑐=(Cl𝜆)^𝑐∨(Cl𝜆^𝑐)^𝑐=Int𝜆^𝑐∨Int𝜆.

The converse of (2) and (3) of Proposition 1 is, in general, not true as is shown by the following example.

Example 1. Let 𝑋={𝑎,𝑏,𝑐} be a set and 𝜏, the fuzzy topology generated by fuzzy sets 𝜇={𝑎_.4,𝑏_.8,𝑐_.2},𝜈={𝑎_.6,𝑏_.9,𝑐_.1}, and 𝜔={𝑎_.5,𝑏_.7,𝑐_.3}. Then, 𝜏={0,𝜇,𝜈,𝜔,{𝑎_.4,𝑏_.8,𝑐_.1},{𝑎_.4,𝑏_.7,𝑐_.2},{𝑎_.5,𝑏_.7,𝑐_.1},{𝑎_.4,𝑏_.7,𝑐_.1},{𝑎_.5,𝑏_.8,𝑐_.3},{𝑎_.5,𝑏_.8,𝑐_.1},{𝑎_.6,𝑏_.9,𝑐_.2},{𝑎_.5,𝑏_.8,𝑐_.2},{𝑎_.6,𝑏_.9,𝑐_.3},{𝑎_.5,𝑏_.7,𝑐_.2},1}. Choose 𝜆={𝑎_.5,𝑏_.7,𝑐_.9} and 𝜓={𝑎_.5,𝑏_.3,𝑐_.3}. Calculations giveBd𝜆=𝑎_.5,𝑏_.3,𝑐_.7𝑎≤𝜆,but𝜆isnotfuzzyclosed,Bd𝜓=_.5,𝑏_.3,𝑐_.7≤𝜓^𝑐,but𝜓isnotfuzzyopen. (6)

The following proposition gives some more properties of fuzzy boundary.

Proposition 2. Let 𝜆 be a fuzzy set in an fts 𝑋. Then, (1)𝐵𝑑𝜆=𝐶𝑙𝜆−𝐼𝑛𝑡𝜆;(2)𝐵𝑑𝐼𝑛𝑡𝜆≤𝐵𝑑𝜆;(3)𝐵𝑑𝐶𝑙𝜆≤𝐵𝑑𝜆;(4)𝐼𝑛𝑡𝜆≤𝜆−𝐵𝑑𝜆.

Proof. (1) Since (Cl𝜆^𝑐)^𝑐=Int𝜆, therefore we have Bd𝜆=Cl𝜆∧Cl𝜆^𝑐=Cl𝜆−(Cl𝜆^𝑐)^𝑐=Cl𝜆−Int𝜆. Thus Bd𝜆=Cl𝜆−Int𝜆. This proves (1),
(2) BdInt𝜆=ClInt𝜆−IntInt𝜆=ClInt𝜆−Int𝜆≤Cl𝜆−Int𝜆=Bd𝜆,
(3) BdCl𝜆=ClCl𝜆−IntCl𝜆=Cl𝜆−IntCl𝜆≤Cl𝜆−Int𝜆=Bd𝜆,
(4)𝜆−Bd𝜆=𝜆∧(Bd𝜆)^𝑐=𝜆∧Cl𝜆∧Cl𝜆^𝑐𝑐=𝜆∧Int𝜆^𝑐=∨Int𝜆𝜆∧Int𝜆^𝑐=∨(𝜆∧Int𝜆)𝜆∧Int𝜆^𝑐∨Int𝜆≥Int𝜆. (7)

The following example shows that the equality does not hold in Proposition 2(2)–(4).

Example 2. Choose 𝛼={𝑎_.1,𝑏_.4,𝑐_.7},𝛽={𝑎_.6,𝑏_.8,𝑐_.5}, and 𝛾={𝑎_.6,𝑏_.3,𝑐_.8} in Example 1. Then, calculations give1≠0𝑎Bd𝛼==BdInt𝛼,Bd𝛽=_.5,𝑏_.2,𝑐_.7≠0𝑎=BdCl𝛽,𝛾−Bd𝛾=_.4,𝑏_.3,𝑐_.2≠Int𝛾=0. (8)

Remark 2. In general topology, the following conditions hold:Bd𝐴∩Int𝐴=𝜙,Int𝐴∪Bd𝐴=Cl𝐴,Int𝐴∪Int𝐴^𝑐∪Bd𝐴=𝑋, (9)whereas, in fuzzy topology, we give counter-examples to show that these may not hold in general.

Example 3. In the fts (𝑋,𝜏) of Example 1, we choose fuzzy set 𝜆={𝑎_.4,𝑏_.7,𝑐_.7}, then calculations giveBd𝜆∧Int𝜆=𝑎_.6,𝑏_.3,𝑐_.8∧𝑎_.4,𝑏_.7,𝑐_.2≠0,Int𝜆∨Bd𝜆=𝑎_.6,𝑏_.7,𝑐_.8≠Cl𝜆=1,Int𝜆∨Int𝜆^𝑐∨Bd𝜆=𝑎_.6,𝑏_.7,𝑐_.8≠1. (10)It is easily seen that Int𝜆∨Bd𝜆≤Cl𝜆.

Theorem 2. Let 𝜆 and 𝜇 be fuzzy sets in an fts 𝑋. Then, 𝐵𝑑(𝜆∨𝜇)≤𝐵𝑑𝜆∨𝐵𝑑𝜇.

Proof. Bd(𝜆∨𝜇)=Cl(𝜆∨𝜇)∧Cl(𝜆∨𝜇)^𝑐≤(Cl𝜆∨Cl𝜇)∧Cl𝜆^𝑐∧Cl𝜇^𝑐=Cl𝜆∧Cl𝜆^𝑐∧Cl𝜇^𝑐∨Cl𝜇∧Cl𝜆^𝑐∧Cl𝜇^𝑐=Bd𝜆∧𝐶𝑙𝜇^𝑐∨Bd𝜇∧Cl𝜆^𝑐≤Bd𝜆∨Bd𝜇. (11)

In Theorem 2, the equality does not hold as is shown by the following:

Example 4. In Example 1, choose fuzzy sets 𝛼={𝑎₀,𝑏₀,𝑐_.2} and 𝛽={𝑎_.4,𝑏_.8,𝑐_.1}. Then, calculations giveBd𝛼∨Bd𝛽=𝑎_.6,𝑏_.2,𝑐_.9≠𝑎_.6,𝑏_.2,𝑐_.8=Bd(𝛼∨𝛽). (12)The following examples show thatBd(𝛾∧𝛿)≰Bd𝛾∧Bd𝛿,Bd𝜓∧Bd𝜉≰Bd(𝜓∧𝜉). (13)For this, choose 𝛾={𝑎_.4,𝑏_.1,𝑐_.9},𝛿={𝑎_.4,𝑏₁,𝑐₁},𝜓={𝑎_.7,𝑏_.4,𝑐_.4}, and 𝜉={𝑎_.4,𝑏_.1,𝑐_.8}. Then, calculations giveBd(𝛾∧𝛿)=𝑎_.4,𝑏_.1,𝑐_.9≰𝑎_.4,𝑏_.1,𝑐_.8𝑎=Bd𝛾∧Bd𝛿,Bd𝜓∧Bd𝜉=_.4,𝑏_.1,𝑐_.8≰𝑎_.4,𝑏_.1,𝑐_.7=Bd(𝜓∧𝜉). (14)However, we have the following theorem.

Theorem 3. For any fuzzy sets 𝜆 and 𝜇 in an fts 𝑋, one has𝐵𝑑(𝜆∧𝜇)≤(𝐵𝑑𝜆∧𝐶𝑙𝜇)∨(𝐵𝑑𝜇∧𝐶𝑙𝜆). (15)

Proof. Bd(𝜆∧𝜇)=Cl(𝜆∧𝜇)∧Cl(𝜆∧𝜇)^𝑐≤(Cl𝜆∧Cl𝜇)∧Cl𝜆^𝑐∨Cl𝜇^𝑐=(Cl𝜆∧Cl𝜇)∧Cl𝜆^𝑐∨(Cl𝜆∧Cl𝜇)∧Cl𝜇^𝑐=(Bd𝜆∧Cl𝜇)∨(Bd𝜇∧Cl𝜆). (16)

Corollary 1. For any fuzzy sets 𝜆 and 𝜇 in an fts 𝑋, one has𝐵𝑑(𝜆∧𝜇)≤𝐵𝑑𝜆∨𝐵𝑑𝜇. (17)

Example 5. To show that the equality in Theorem 3, in general, does not hold, choose 𝜆={𝑎_.6,𝑏_.6,𝑐_.7} and 𝜇={𝑎_.6,𝑏_.2,𝑐_.9} in the fts 𝑋 defined in Example 1. Then, calculations give(Bd𝜆∧Cl𝜇)∨(Bd𝜇∧Cl𝜆)=𝜇≰𝑎_.6,𝑏_.2,𝑐_.8=Bd(𝜆∧𝜇). (18)

In general topology, it is known thatBdBdBd𝐴=BdBd𝐴, (19)for any subset 𝐴 of a space 𝑋. However, in fuzzy topology, we have the following proposition.

Proposition 3. For any fuzzy set 𝜆 in an fts 𝑋, one has (1)𝐵𝑑𝐵𝑑𝜆≤𝐵𝑑𝜆;(2)𝐵𝑑𝐵𝑑𝐵𝑑𝜆≤𝐵𝑑𝐵𝑑𝜆.

Proof. (1) BdBd𝜆=ClBd𝜆∧Cl(Bd𝜆)^𝑐≤ClBd𝜆=Bd𝜆.
(2)BdBdBd𝜆=ClBdBd𝜆∧Cl(BdBd𝜆)^𝑐=BdBd𝜆∧Cl(BdBd𝜆)^𝑐≤BdBd𝜆. (20)

Remark 3. We could not find an example to show that the equality in (2) does not hold . However, the equality in (1), in general, does not hold as is shown by the following example.

Example 6. Choose 𝜆={𝑎_.2,𝑏_.5,𝑐_.9} in fts 𝑋 of Example 1. Then, Bd𝜆=1 but BdBd𝜆=0.

Definition 7 (see [27]). If 𝜆 is a fuzzy set of 𝑋 and 𝜇 is a fuzzy set of 𝑌, then(𝜆×𝜇)(𝑥,𝑦)=min𝜆(𝑥),𝜇(𝑦),foreach(𝑥,𝑦)∈𝑋×𝑌. (21)

Definition 8 (see [13]). An fts (𝑋,𝜏_𝑋) is product related to another fts (𝑌,𝜏_𝑌) if for any fuzzy set 𝜈 of 𝑋 and 𝜉 of 𝑌 whenever 𝜆^𝑐≱𝜈 and 𝜇^𝑐≱𝜉 imply 𝜆^𝑐×1∨1×𝜇^𝑐≥𝜈×𝜉, where 𝜆∈𝜏_𝑋 and 𝜇∈𝜏_𝑌, there exist 𝜆₁∈𝜏_𝑋 and 𝜇₁∈𝜏_𝑌 such that 𝜆^𝑐₁≥𝜈 and 𝜇^𝑐₁≥𝜉 and 𝜆^𝑐₁×1∨1×𝜇^𝑐₁=𝜆^𝑐×1∨1×𝜇^𝑐.

Theorem 4 (see [13]). Let 𝑋 and 𝑌 be product-related fts's. Then, for a fuzzy set 𝜆 of 𝑋 and a fuzzy set 𝜇 of 𝑌, one has (1)𝐶𝑙(𝜆×𝜇)=𝐶𝑙𝜆×𝐶𝑙𝜇,(2)𝐼𝑛𝑡(𝜆×𝜇)=𝐼𝑛𝑡𝜆×𝐼𝑛𝑡𝜇.

Lemma 2. For fuzzy sets 𝜆, 𝜇, 𝜈, and 𝜔 in a set 𝑋, one has(𝜆∧𝜇)×(𝜈∧𝜔)=(𝜆×𝜔)∧(𝜇×𝜈). (22)

Proof. =(𝜆∧𝜇)×(𝜈∧𝜔)(𝑥,𝑦)=min(𝜆∧𝜇)(𝑥),(𝜇∧𝜔)(𝑦)=minmin𝜆(𝑥),𝜇(𝑥),min𝜈(𝑦),𝜔(𝑦)=minmin𝜆(𝑥),𝜔(𝑦),min𝜇(𝑥),𝜈(𝑦)=min(𝜆×𝜔)(𝑥,𝑦),(𝜇×𝜈)(𝑥,𝑦)(𝜆×𝜔)∧(𝜇×𝜔)(𝑥,𝑦). (23)

Theorem 5. Let 𝑋_𝑖,𝑖=1,2,…,𝑛, be a family of product-related fuzzy topological spaces. If each 𝜆_𝑖 is a fuzzy set in 𝑋_𝑖, then𝐵𝑑^𝑛_𝑖=1𝜆_𝑖=𝐵𝑑𝜆₁×𝐶𝑙𝜆₂×⋯×𝐶𝑙𝜆_𝑛∨𝐶𝑙𝜆₁×𝐵𝑑𝜆₂×𝐶𝑙𝜆₃×⋯×𝐶𝑙𝜆_𝑛∨⋯∨𝐶𝑙𝜆₁×𝐶𝑙𝜆₂×⋯×𝐵𝑑𝜆_𝑛. (24)

Proof. We use Proposition 2(1), Theorem 4, and Lemma 2 to prove this.
It suffices to prove this for 𝑛=2. Consider Bd𝜆₁×𝜆₂=Cl𝜆₁×𝜆₂−Int𝜆₁×𝜆₂=Cl𝜆₁×Cl𝜆₂−Int𝜆₁×Int𝜆₂=Cl𝜆₁×Cl𝜆₂−Int𝜆₁∧Cl𝜆₁×Int𝜆₂∧Cl𝜆₂=Cl𝜆₁×Cl𝜆₂−Int𝜆₁×Cl𝜆₂∧Cl𝜆₁×Int𝜆₂=(byLemma2)Cl𝜆₁×Cl𝜆₂−Int𝜆₁×Cl𝜆₂∨Cl𝜆₁×Cl𝜆₂−Cl𝜆₁×Int𝜆₂=Cl𝜆₁−Int𝜆₁×Cl𝜆₂∨Cl𝜆₁×Cl𝜆₂−Int𝜆₂=Bd𝜆₁×Cl𝜆₂∨Cl𝜆₁×Bd𝜆₂. (25)

Theorem 6. Let 𝑓∶𝑋→𝑌 be a fuzzy continuous function. Then,𝐵𝑑𝑓⁻¹(𝜇)≤𝑓⁻¹(𝐵𝑑𝜇), (26)for any fuzzy set 𝜇 in 𝑌.

Proof. Let 𝑓 be fuzzy continuous and 𝜇 a fuzzy set in 𝑌. Then,
Bd𝑓⁻¹(𝜇)=Cl𝑓⁻¹(𝜇)∧Cl𝑓⁻¹(𝜇)^𝑐≤Cl𝑓⁻¹(Cl𝜇)∧Cl𝑓⁻¹Cl𝜇^𝑐=𝑓⁻¹(Cl𝜇)∧𝑓⁻¹Cl𝜇^𝑐=𝑓⁻¹Cl𝜇∧Cl𝜇^𝑐=𝑓⁻¹(Bd𝜇). (27)Therefore, Bd𝑓⁻¹(𝜇)≤𝑓⁻¹(Bd𝜇).

Before closing this section, we give an interesting characterization of fuzzy continuous functions in terms of fuzzy-derived set and fuzzy closure. It shows that fuzzy continuity, in essence, amounts to preservation of fuzzy closedness. For this we first recall following definitions.

Definition 9 (see [24]). A fuzzy set in 𝑋 is called a fuzzy point if and only if it takes the value 0 for all 𝑥∈𝑋 except one, say, 𝑒∈𝑋. If its value at 𝑒 is 𝛼(0<𝛼≤1), one denotes this fuzzy point by 𝑒_𝛼. The point 𝑒∈𝑋 is called support of fuzzy point 𝑒_𝛼 and is denoted as supp(𝑒).

Definition 10 (see [24]). A fuzzy point 𝑒_𝛼 is said to be quasicoincident (also written as 𝑞-coincident or 𝑄-coincident) with a fuzzy set 𝜆, denoted by 𝑒_𝛼𝑞𝜆, if for some 𝑥∈𝑋,𝛼>𝜆^𝑐(𝑥), or 𝛼+𝜆(𝑥)>1.

Definition 11 (see [24]). A fuzzy set 𝜆 in an fts 𝑋 is called a 𝑄-neighborhood of a fuzzy point 𝑒, if there exists a 𝜇∈𝜏 such that 𝑒𝑞𝜇≤𝜆.

Definition 12 (see [24]). A fuzzy point 𝑒 is called an adherent point of a fuzzy set 𝜆, if every 𝑄-neighborhood of 𝑒 is quasicoincident with 𝜆.

Definition 13 (see [24]). A fuzzy point 𝑒 is called an accumulation point of a fuzzy set 𝜆, if 𝑒 is an adherent point of 𝜆 and every 𝑄-neighborhood of 𝑒 and 𝜆 is quasicoincident at some point different from supp(𝑒), whenever 𝑒∈𝜆. The union of all the accumulation points of 𝜆 is called the derived set of 𝜆, denoted as 𝜆^𝑑. It is evident that 𝜆^𝑑≤Cl𝜆.

Proposition 4 (see [24]). Let 𝜆 be a fuzzy set in an fts 𝑋, then (1)𝐶𝑙𝜆=𝜆∨𝜆^𝑑;(2)𝜆 is fuzzy closed if and only if 𝜆^𝑑≤𝜆.

We use Proposition 4 and prove the following theorem.

Theorem 7. Let 𝑓∶𝑋→𝑌 be a function. Then, the following conditions are equivalent: (1)𝑓∶𝑋→𝑌 is fuzzy continuous;(2)𝑓(𝜆^𝑑)≤𝐶𝑙𝑓(𝜆), for any fuzzy set 𝜆 in 𝑋.

Proof. (1)⇒(2) Let 𝑓 be fuzzy continuous and 𝜆 a fuzzy set in 𝑋. Since Cl𝑓(𝜆) is fuzzy closed in 𝑌, then 𝑓⁻¹(Cl𝑓(𝜆)) is fuzzy closed in 𝑋. 𝜆≤𝑓⁻¹(Cl𝑓(𝜆)) gives Cl𝜆≤Cl𝑓⁻¹(Cl𝑓(𝜆))=𝑓⁻¹(Cl𝑓(𝜆)). Therefore, 𝑓(𝜆^𝑑)≤𝑓(Cl𝜆)≤𝑓𝑓⁻¹(Cl𝑓(𝜆))≤Cl𝑓(𝜆). Consequently, 𝑓(𝜆^𝑑)≤Cl𝑓(𝜆).
(2)⇒(1) Suppose 𝑓(𝜆^𝑑)≤Cl𝑓(𝜆),𝜆∈𝐼^𝑋. Let 𝜇 be any fuzzy closed set in 𝑌. We show that 𝑓⁻¹(𝜇) is fuzzy closed in 𝑋. By our hypothesis, 𝑓([𝑓⁻¹(𝜇)]^𝑑)≤Cl𝑓(𝑓⁻¹(𝜇))≤Cl𝜇=𝜇,𝑓([𝑓⁻¹(𝜇)]^𝑑)≤𝜇 gives [𝑓⁻¹(𝜇)]^𝑑≤𝑓⁻¹(𝑓[𝑓⁻¹(𝜇)]^𝑑)≤𝑓⁻¹(𝜇), or [𝑓⁻¹(𝜇)]^𝑑≤𝑓⁻¹(𝜇) implies that 𝑓⁻¹(𝜇) is fuzzy closed in 𝑋. Thus, 𝑓 is fuzzy continuous.

4. Fuzzy Semiboundary

First, we recall some notions.

Definition 14 (see [13]). Let 𝜆 be a fuzzy set in an fts (𝑋,𝜏). Then, 𝜆 is called a fuzzy semiopen set of 𝑋 if there exists a 𝜈∈𝜏 such that 𝜈≤𝜆≤Cl𝜈.

Definition 15 (see [22]). Let 𝜆 be a fuzzy set in an fts 𝑋. Then, semiclosure (briefly 𝑠Cl) and semi-interior (briefly 𝑠Int) of 𝜆 are given as𝑠Cl𝜆=∧{𝛽∣𝜆≤𝛽,𝛽fuzzysemiclosed},𝑠Int𝜆=∨{𝛽∣𝛽≤𝜆,𝛽fuzzysemiopen}. (28)

Remark 4. In the following theorems, we note that almost all the properties related to fuzzy semi-interior, fuzzy semiclosure, and fuzzy semiboundary are analogous to their counterparts in fuzzy topology and hence some of the proofs are not given.

Theorem 8. For fuzzy sets 𝜆 and 𝜇 in an fts 𝑋, one has (1)𝑠𝐼𝑛𝑡(𝜆∨𝜇)≥𝑠𝐼𝑛𝑡𝜆∨𝑠𝐼𝑛𝑡𝜇;(2)𝑠𝐼𝑛𝑡(𝜆∧𝜇)=𝑠𝐼𝑛𝑡𝜆∧𝑠𝐼𝑛𝑡𝜇;(3)𝑠𝐶𝑙(𝜆∨𝜇)=𝑠𝐶𝑙𝜆∨𝑠𝐶𝑙𝜇;(4)𝑠𝐶𝑙(𝜆∧𝜇)≤𝑠𝐶𝑙𝜆∧𝑠𝐶𝑙𝜇;(5)𝑠𝐶𝑙𝑠𝐶𝑙𝜆=𝑠𝐶𝑙𝜆;(6)𝑠𝐼𝑛𝑡𝑠𝐼𝑛𝑡𝜆=𝑠𝐼𝑛𝑡𝜆;(7)(𝑠𝐼𝑛𝑡𝜆)^𝑐=𝑠𝐶𝑙(𝜆^𝑐);(8)(𝑠𝐶𝑙𝜆)^𝑐=𝑠𝐼𝑛𝑡(𝜆^𝑐).

Proof. (1) 𝑠Int𝜆 and 𝑠Int𝜇 are both fuzzy semiopen and 𝜆≤𝜆∨𝜇, 𝜇≤𝜆∨𝜇 imply 𝑠Int𝜆≤𝑠Int(𝜆∨𝜇) and 𝑠Int𝜇≤𝑠Int(𝜆∨𝜇). Combining, 𝑠Int𝜆∨𝑠Int𝜇≤𝑠Int(𝜆∨𝜇) or𝑠Int(𝜆∨𝜇)≥𝑠Int𝜆∨𝑠Int𝜇. (29)
(2) 𝜆∧𝜇≤𝜆 and 𝜆∧𝜇≤𝜇 imply 𝑠Int(𝜆∧𝜇)≤𝑠Int𝜆,𝑠Int(𝜆∧𝜇)≤𝑠Int𝑡𝜇 and therefore 𝑠Int(𝜆∧𝜇)≤𝑠Int𝜆∧𝑠Int𝜇. Conversely 𝑠Int𝜆≤𝜆 and 𝑠Int𝜇≤𝜇 implies 𝑠Int𝜆∧𝑠Int𝜇≤𝜆∧𝜇 and 𝑠Int𝜆∧𝑠Int𝜇 is fuzzy semiopen. But 𝑠Int(𝜆∧𝜇) is the largest fuzzy semiopen set contained in 𝜆∧𝜇, hence 𝑠Int∧𝑠Int𝜇≤𝑠Int(𝜆∧𝜇). This gives the equality.
(3) It follows easily from (2).
(4) Since 𝜆∧𝜇≤𝜆,𝜆∧𝜇≤𝜇⟹𝑠Cl(𝜆∧𝜇)≤𝑠Cl𝜆,𝑠Cl(𝜆∧𝜇)≤𝑠Cl𝜇⟹𝑠Cl(𝜆∧𝜇)≤𝑠Cl𝜆∧𝑠Cl𝜇. (30)
(5)–(8) Proofs are straightforward.

The inequalities (1) and (4) of Theorem 8 are irreversible as is shown by the following example.

Example 7. Let 𝑋={𝑎,𝑏} be a set and 𝐼={0,.5,1} the lattice of membership grades for fuzzy sets in 𝑋. Let 𝜏 be the fuzzy topology on 𝑋 given as𝜏=0,𝑎_.5,𝑏₁,𝑎_.0,𝑏₁,𝑎₁,𝑏_.5,𝑎_.5,𝑏_.5,𝑎₀,𝑏_.5,1. (31)We choose fuzzy sets 𝛼={𝑎_.5,𝑏₀},𝛽={𝑎₀,𝑏_.5},𝛾={𝑎_.5,𝑏₁}, and 𝛿={𝑎₁,𝑏_.5}. Then, calculations give𝑠Int(𝛼∨𝛽)=𝑎_.5,𝑏_.5≰𝑎₀,𝑏_.5𝑎=𝑠Int𝛼∨𝑠Int𝛽,𝑠Cl𝛾∧𝑠Cl𝛿=₁,𝑏_.5≰𝑎_.5,𝑏_.5=𝑠Cl(𝛼∧𝛿). (32)

Definition 16 (see [25]). Let 𝜆 be a fuzzy set in an fts 𝑋. Then, the fuzzy semiboundary of 𝜆 is defined as 𝑠Bd𝜆=𝑠Cl𝜆∧𝑠Cl𝜆^𝑐. Obviously, 𝑠Bd𝜆 is a fuzzy semiclosed set.

Remark 5. In fuzzy topology, we have 𝜆∨𝑠Bd𝜆≤𝑠Cl𝜆, for an arbitrary fuzzy set 𝜆 in 𝑋, the equality does not hold as the following example shows.

Example 8. Let 𝑋={𝑎,𝑏} be a set with fuzzy topology 𝜏={0,{𝑎_.4,𝑏_.8},{𝑎_.6,𝑏_.9},{𝑎_.5,𝑏_.7},{𝑎_.5,𝑏_.2},{𝑎_.8,𝑏_.7},{𝑎_.3,𝑏_.2},{𝑎_.4,𝑏_.7},{𝑎_.4,𝑏_.2},{𝑎_.6,𝑏_.7},{𝑎_.5,𝑏_.8},{𝑎_.8,𝑏_.8},{𝑎_.6,𝑏_.8},{𝑎_.8,𝑏_.9},1}. Choose 𝜆={𝑎_.4,𝑏_.7}, then calculations give𝑠Cl𝜆=𝑎_.5,𝑏_.8≠𝑎_.5,𝑏_.7=𝜆∨𝑠Bd𝜆. (33)

In the following theorem, (1)–(5) are analogs of Proposition 1 and hence we omit their proofs.

Proposition 5. For a fuzzy set 𝜆 in an fts 𝑋, the following conditions hold. (1)𝑠𝐵𝑑𝜆=𝑠𝐵𝑑𝜆^𝑐.(2)If 𝜆 is fuzzy semiclosed, then 𝑠𝐵𝑑𝜆≤𝜆.(3)If 𝜆 is fuzzy semiopen, then 𝑠𝐵𝑑𝜆≤𝜆^𝑐.(4)Let 𝜆≤𝜇 and 𝜇∈𝐹𝑆𝐶(𝑋) (resp., 𝜇∈𝐹𝑆𝑂(𝑋)) . Then, 𝑠𝐵𝑑𝜆≤𝜇 (resp., 𝑠𝐵𝑑𝜆≤𝜇^𝑐), where 𝐹𝑆𝐶(𝑋) (resp., 𝐹𝑆𝑂(𝑋)) denotes the class of fuzzy semiclosed (resp., fuzzy semiopen) sets in 𝑋.(5)(𝑠𝐵𝑑𝜆)^𝑐=𝑠𝐼𝑛𝑡𝜆∨𝑠𝐼𝑛𝑡𝜆^𝑐.(6)𝑠𝐵𝑑𝜆≤𝐵𝑑𝜆.(7)𝑠𝐶𝑙𝑠𝐵𝑑𝜆≤𝐵𝑑𝜆.

Proof. (6) Since 𝑠Cl𝜆≤Cl𝜆 and 𝑠Cl𝜆^𝑐≤Cl𝜆^𝑐, then we have𝑠Bd𝜆=𝑠Cl𝜆∧𝑠Cl𝜆^𝑐≤Cl𝜆∧Cl𝜆^𝑐=Bd𝜆. (34)
(7)𝑠Cl𝑠Bd𝜆=𝑠Cl𝑠Cl𝜆∧𝑠Cl𝜆^𝑐≤𝑠Cl𝑠Cl𝜆∧𝑠Cl𝑠Cl𝜆^𝑐=𝑠Cl𝜆∧𝑠Cl𝜆^𝑐=𝑠Bd𝜆≤Bd𝜆. (35)

The converse of (2) and (3) and reverse inequalities of (6) and (7) in Proposition 5 are, in general, not true as is shown by the following example.

Example 9. Choose 𝜆={𝑎_.7,𝑏_.7}, 𝜓={𝑎_.3,𝑏_.3},𝜔={𝑎_.7,𝑏_.9}, 𝜉={𝑎_.6,𝑏₁} in the fts (𝑋,𝜏) defined in Example 8. Then, calculations give𝑠Bd𝜆=𝑎_.4,𝑏_.3𝑎≤𝜆,but𝜆isnotfuzzysemiclosed,𝑠Bd𝜓=_.4,𝑏_.3≤𝜓^𝑐𝑎,but𝜓isnotfuzzysemiopen,Bd𝜔=_.4,𝑏_.1≰𝑎_.3,𝑏_.1𝑎=𝑠Bd𝜔,Bd𝜉=_.4,𝑏_.1≰𝑎_.4,𝑏₀=𝑠Cl𝑠Bd𝜉. (36)

The following is analog of Proposition 2 and hence we omit its proof.

Proposition 6. Let 𝜆 be a fuzzy set in an fts 𝑋. Then, one has (1)𝑠𝐵𝑑𝜆=𝑠𝐶𝑙𝜆−𝑠𝐼𝑛𝑡𝜆,(2)𝑠𝐵𝑑𝑠𝐼𝑛𝑡𝜆≤𝑠𝐵𝑑𝜆,(3)𝑠𝐵𝑑𝑠𝐶𝑙𝜆≤𝑠𝐵𝑑𝜆,(4)𝑠𝐼𝑛𝑡𝜆≤𝜆−𝑠𝐵𝑑𝜆.

To show that the inequalities (2), (3), and (4) of Proposition 6, are, in general, irreversible, we have the following example.

Example 10. Choose 𝛼={𝑎_.7,𝑏_.5} and 𝜇={𝑎_.6,𝑏_.3} in the fts 𝑋 defined in Example 8. Then, calculations give𝑠Bd𝛼=𝑎_.5,𝑏_.8≰𝑎_.5,𝑏_.2𝑎=𝑠Bd𝑠Int𝛼,𝑠Bd𝛼=_.5,𝑏_.8≰𝑎_.4,𝑏_.2𝑎=𝑠Bd𝑠Cl𝛼,𝜇−𝑠Bd𝜇=_.5,𝑏_.3≰𝑎_.5,𝑏_.2=𝑠Int𝜇. (37)

Remark 6. In general topology, the following conditions hold:𝑠Bd𝐴∩𝑠Int𝐴=𝜙,𝑠Int𝐴∪𝑠Bd𝐴=Cl𝐴,𝑠Int𝐴∪𝑠Int𝐴^𝑐∪𝑠Bd𝐴=𝑋, (38)whereas, in fuzzy topology, we give counter-examples to show that these may not hold in general.

Example 11. In the fts 𝑋 of Example 8, we choose fuzzy set 𝜆={𝑎_.7,𝑏_.5}, then calculations give𝑠Bd𝜆∧𝑠Int𝜆=𝑎_.5,𝑏_.8∧𝑎_.5,𝑏_.2≠0,𝑠Int𝜆∨𝑠Bd𝜆=𝑎_.5,𝑏_.8≠𝑎_.7,𝑏_.8=𝑠Cl𝜆,𝑠Int𝜆∨𝑠Int𝜆^𝑐∨𝑠Bd𝜆=𝑎_.5,𝑏_.8≠1. (39)It is easily seen that 𝑠Int𝜆∨𝑠Bd𝜆≤𝑠Cl𝜆.

Theorem 9. Let 𝜆 and 𝜇 be fuzzy sets in an fts 𝑋. Then, 𝑠𝐵𝑑(𝜆∨𝜇)≤𝑠𝐵𝑑𝜆∨𝑠𝐵𝑑𝜇.

The reverse inequality in Theorem 9 is, in general, not true as is shown by the following example.

Example 12. In Example 8, choose fuzzy sets 𝛼={𝑎_.4,𝑏_.7} and 𝛽={𝑎_.6,𝑏_.2}. Then, calculations give𝑠Bd𝛼∨𝑠Bd𝛽=𝑎_.5,𝑏_.3≰𝑎_.4,𝑏_.3=𝑠Bd(𝛼∨𝛽). (40)The following example shows that𝑠Bd(𝛾∧𝛿)≰𝑠Bd𝛾∧𝑠Bd𝛿,𝑠Bd𝜆∧𝑠Bd𝛿≰𝑠Bd(𝛾∧𝛿). (41)For this choose 𝛾={𝑎_.4,𝑏_.8} and 𝛿={𝑎_.6,𝑏_.3}. Then, calculations give𝑠Bd(𝛾∧𝛿)=𝑎_.4,𝑏_.3≰𝑎_.5,𝑏_.2𝑎=𝑠Bd𝛾∧𝑠Bd𝛿,𝑠Bd𝜆∧𝑠Bd𝛿=_.5,𝑏_.2≰𝑎_.4,𝑏_.3=𝑠Bd(𝛾∧𝛿), (42)However, we have the following theorem which is an analog of Theorem 3.

Theorem 10. For any fuzzy sets 𝜆 and 𝜇 in an fts 𝑋, one has𝑠𝐵𝑑(𝜆∧𝜇)≤(𝑠𝐵𝑑𝜆∧𝑠𝐶𝑙𝜇)∨(𝑠𝐵𝑑𝜇∧𝑠𝐶𝑙𝜆). (43)

Corollary 2. For any fuzzy sets 𝜆 and 𝜇 in an fts 𝑋, one has𝑠𝐵𝑑(𝜆∧𝜇)≤𝑠𝐵𝑑𝜆∨𝑠𝐵𝑑𝜇. (44)

Example 13. To show that the reverse inequality in Theorem 10 is, in general, not true, choose fuzzy sets 𝛾 and 𝛿 as given in Example 8. Then, calculations give(𝑠Bd𝛾∧𝑠Cl𝛿)∨(𝑠Bd𝛾∧𝑠Cl𝛿)=𝑎_.5,𝑏_.3≰𝑎_.4,𝑏_.3=𝑠Bd(𝛾∧𝛿). (45)

The analog of Proposition 3 is the following theorem, the proof of which is similar.

Proposition 7. For any fuzzy set 𝜆 in an fts 𝑋, one has (1)𝑠𝐵𝑑𝑠𝐵𝑑𝜆≤𝑠𝐵𝑑𝜆,(2)𝑠𝐵𝑑𝑠𝐵𝑑𝑠𝐵𝑑𝜆≤𝑠𝐵𝑑𝑠𝐵𝑑𝜆.

Remark 7. As in the case of Proposition 3(2), we also do not know whether the equality in Proposition 7(2) holds or not. However, the reverse inequality of (1) is, in general, not true as is shown by the following example.

Example 14. Choose 𝜆={𝑎_.6,𝑏_.4} in fts 𝑋 of Example 8. Then,𝑠Bd𝜆=𝑎_.5,𝑏_.8≰𝑎_.5,𝑏_.2=𝑠Bd𝑠Bd𝜆. (46)

Lemma 3 (see [13]). If 𝜆 is a fuzzy set of 𝑋 and 𝜇 is a fuzzy set of 𝑌, then1−(𝜆×𝜇)=𝜆^𝑐×1∨1×𝜇^𝑐. (47)

Using Lemma 3, we have the following one.

Lemma 4. Let 𝜆 be a fuzzy semiclosed (resp., fuzzy semiopen [13]) set of an fts 𝑋 and 𝜇 a fuzzy semiclosed (resp., fuzzy semiopen) set of an fts 𝑌. Then, 𝜆×𝜇 is a fuzzy closed (resp., fuzzy semiopen) set of the fuzzy product space 𝑋×𝑌.

Using Lemma 4, we have the following theorem.

Theorem 11. If 𝜆 is a fuzzy set of fts 𝑋 and 𝜇 of fts 𝑌, then (1)𝑠𝐶𝑙𝜆×𝑠𝐶𝑙𝜇≥𝑠𝐶𝑙(𝜆×𝜇), (2)𝑠𝐼𝑛𝑡𝜆×𝑠𝐼𝑛𝑡𝜇≤𝑠𝐼𝑛𝑡(𝜆×𝜇).

Moreover, we have the following one.

Theorem 12. Let 𝑋 and 𝑌 be product-related fts's. Then, for a fuzzy set 𝜆 of 𝑋 and a fuzzy set 𝜇 of 𝑌, one has (1)𝑠𝐶𝑙(𝜆×𝜇)=𝑠𝐶𝑙𝜆×𝑠𝐶𝑙𝜇,(2)𝑠𝐼𝑛𝑡(𝜆×𝜇)=𝑠𝐼𝑛𝑡𝜆×𝑠𝐼𝑛𝑡𝜇.

Proof. (1) For fuzzy sets 𝜆^′_𝑖𝑠 of 𝑋 and 𝜇^′_𝑗𝑠 of 𝑌, we first note that (i)inf {𝜆_𝑖,𝜇_𝑗}=min{inf𝜆_𝑖,inf𝜇_𝑗},(ii)inf {𝜆_𝑖×1}=(inf𝜆_𝑖)×1,(iii)inf {1×𝜇_𝑗}=1×(inf𝜇_𝑗). In view of Theorem 11, it is sufficient to show that 𝑠Cl(𝜆×𝜇)≥𝑠Cl𝜆×𝑠Cl𝜇. Let 𝜆_𝑖∈FSO(𝑋) and 𝜇_𝑗∈FSO(𝑌). Then,𝑠Cl(𝜆×𝜇)=inf𝜆_𝑖×𝜇_𝑗^𝑐∣𝜆_𝑖×𝜇_𝑗^𝑐𝜆≥𝜆×𝜇=inf^𝑐_𝑖×1∨1×𝜇^𝑐_𝑗∣𝜆^𝑐_𝑖×1∨1×𝜇^𝑐_𝑗𝜆≥𝜆×𝜇=inf^𝑐_𝑖×1∨1×𝜇^𝑐_𝑗∣𝜆^𝑐_𝑖≥𝜆or𝜇^𝑐_𝑗𝜆≥𝜇=mininf^𝑐_𝑖×1∨1×𝜇^𝑐_𝑗∣𝜆^𝑐_𝑖,𝜆≥𝜆inf^𝑐_𝑖×1∨1×𝜇^𝑐_𝑗∣𝜇^𝑐_𝑗≥𝜇. (48)Sinceinf𝜆^𝑐_𝑖×1∨1×𝜇^𝑐_𝑗∣𝜆^𝑐_𝑖≥𝜆≥inf𝜆^𝑐_𝑖×1∣𝜆^𝑐_𝑖𝜆≥𝜆=inf^𝑐_𝑖∣𝜆^𝑐_𝑖×11,𝜆≥𝜆=(𝑠Cl𝜆)×inf^𝑐_𝑖×1∨1×𝜇^𝑐_𝑗∣𝜇^𝑐_𝑗≥𝜇≥inf𝜇^𝑐_𝑗×1∣𝜇^𝑐_𝑗=1𝜇≥𝜇×inf^𝑐_𝑗∣𝜇^𝑐_𝑗=1≥𝜇×(𝑠Cl𝜇), (49)we have𝑠Cl(𝜆×𝜇)≥min(𝑠Cl𝜆×1,1×𝑠Cl𝜇)=𝑠Cl𝜆×𝑠Cl𝜇. (50)
(2) This follows from (1) using the facts that (𝑠Int𝜓)^𝑐=𝑠Cl𝜓^𝑐 and (𝑠Cl𝜓)^𝑐=𝑠Int𝜓^𝑐.

The analog of Theorem 5 is the following, the proof of which is similar.

Theorem 13. Let 𝑋_𝑖,𝑖=1,2,…,𝑛, be a family of product-related fuzzy topological spaces. If each 𝜆_𝑖 is a fuzzy set in 𝑋_𝑖, then𝑠𝐵𝑑^𝑛_𝑖=1𝜆_𝑖=𝑠𝐵𝑑𝜆₁×𝑠𝐶𝑙𝜆₂×⋯×𝑠𝐶𝑙𝜆_𝑛∨𝑠𝐶𝑙𝜆₁×𝑠𝐵𝑑𝜆₂×𝑠𝐶𝑙𝜆₃×⋯×𝑠𝐶𝑙𝜆_𝑛∨⋯∨𝑠𝐶𝑙𝜆₁×𝑠𝐶𝑙𝜆₂×⋯×𝑠𝐵𝑑𝜆_𝑛. (51)

The following theorem gives a necessary condition for fuzzy semicontinuous functions in terms of fuzzy boundary and fuzzy semiboundary.

Theorem 14. Let 𝑓∶𝑋→𝑌 be a fuzzy semicontinuous function. Then, one has𝑠𝐵𝑑𝑓⁻¹(𝜇)≤𝑓⁻¹(𝐵𝑑𝜇), (52)for any fuzzy set 𝜇 in 𝑌.

Proof. Let 𝑓 be fuzzy semicontinuous and 𝜇 a fuzzy set in 𝑌. Then, Cl𝜇 is fuzzy closed in 𝑌, which implies that 𝑓⁻¹(Cl𝜇) is fuzzy semiclosed in 𝑋. Therefore,
𝑠Bd𝑓⁻¹(𝜇)=𝑠Cl𝑓⁻¹(𝜇)∧𝑠Cl𝑓⁻¹(𝜇)^𝑐≤𝑠Cl𝑓⁻¹(Cl𝜇)∧𝑠Cl𝑓⁻¹Cl𝜇^𝑐=𝑓⁻¹(Cl𝜇)∧𝑓⁻¹Cl𝜇^𝑐=𝑓⁻¹Cl𝜇∧Cl𝜇^𝑐=𝑓⁻¹(Bd𝜇). (53)Hence, 𝑠Bd𝑓⁻¹(𝜇)≤𝑓⁻¹(Bd𝜇).

Definition 17 (see [28]). A function 𝑓∶𝑋→𝑌 is said to be fuzzy irresolute if 𝑓⁻¹(𝛽) is fuzzy semiopen in 𝑋, for each fuzzy semiopen set 𝛽 in 𝑌.

The following theorem gives a necessary condition of fuzzy irresolute functions in terms of fuzzy boundary and fuzzy semiboundary, the proof of which is similar to Theorem 14.

Theorem 15. Let 𝑓∶𝑋→𝑌 be a fuzzy irresolute function. Then, one has𝑠𝐵𝑑𝑓⁻¹(𝜇)≤𝑓⁻¹(𝑠𝐵𝑑𝜇), (54)for any fuzzy set 𝜇 in 𝑌.

Definition 18. A fuzzy set 𝜆 in an fts 𝑋 is called a fuzzy semi-𝑄-neighborhood of a fuzzy point 𝑒, if there exists a fuzzy semiopen set 𝜇 in 𝑋, such that 𝑒𝑞𝜇≤𝜆.

Theorem 16. A fuzzy point 𝑒=𝑥_𝛼∈𝑠𝐶𝑙𝜆 if each semi-𝑄-neighborhood of 𝑒 is quasicoincident with 𝜆.

Proof. 𝑥_𝛼∈𝑠Cl𝜆 if and only if for every fuzzy closed set 𝜓≥𝜆,𝑥_𝛼∈𝜓. This gives 𝜓(𝑥_𝛼)≥𝜆(𝑥_𝛼). Equivalently, 𝑥_𝛼∈𝑠Cl𝜆 if and only if for every fuzzy semiopen set 𝛽≤𝜆^𝑐,𝛽(𝑥_𝛼)≤𝜆^𝑐(𝑥_𝛼). That is, for every fuzzy open set 𝛽 satisfying 𝛽(𝑥_𝛼)≥𝜆^𝑐,𝛽 is not contained in 𝜆^𝑐, or 𝛽𝑞𝜆^𝑐𝑐=𝜆. Thus, 𝑥_𝛼∈𝑠Cl𝜆 if every fuzzy open 𝑄-neighborhood 𝛽 of 𝑥_𝛼 is quasicoincident with 𝜆.

Definition 19. A fuzzy point 𝑒 is called semiadherent point of a fuzzy set 𝜆 if every semi-𝑄-neighborhood of 𝑒 is quasicoincident with 𝜆.

Definition 20. A fuzzy point 𝑒 is called a semiaccumulation point of a fuzzy set 𝜆 if 𝑒 is a semiadherent point of 𝜆 and every semi-𝑄-neighborhood of 𝑒 and 𝜆 is quasicoincident at some point different from supp (𝑒), whenever 𝑒∈𝜆. The union of all the semiaccumulation points of 𝜆 is called the fuzzy semiderived set of 𝜆, denoted as 𝜆^sd. It is evident that 𝜆^sd≤𝑠Cl𝜆.

Proposition 8. Let 𝜆 be a fuzzy set in 𝑋, then 𝑠𝐶𝑙𝜆=𝜆∨𝜆^𝑠𝑑.

Proof. Let Ω={𝑒∣𝑒isasemiadherentpointof𝜆}. Then, from Theorem 16, 𝑠𝐶𝑙𝜆=∨Ω. On the other hand, 𝑒∈Ω is either 𝑒∈𝜆 or 𝑒∉𝜆; for the latter case, by Definition 20, 𝑒∈𝜆^sd, hence 𝑠𝐶𝑙𝜆=∨Ω≤𝜆∨𝜆^sd. The reverse inclusion is obvious.

Corollary 3. For any fuzzy set 𝜆 in an fts 𝑋,𝜆 is fuzzy semiclosed if and only if 𝜆^𝑠𝑑≤𝜆.

Definition 21 (see [13]). Let 𝑓∶𝑋→𝑌 be a function from an fts 𝑋 to another fts 𝑌. Then, 𝑓 is said to be fuzzy semicontinuous function if 𝑓⁻¹(𝜆) is fuzzy semiopen in 𝑋, for each fuzzy open set 𝜆 in 𝑌.

We use Corollary 3 and characterize fuzzy semicontinuous functions in terms of fuzzy semiderived set as follows.

Theorem 17. Let 𝑓∶𝑋→𝑌 be a function. Then, the following conditions are equivalent: (1)𝑓 is fuzzy semicontinuous,(2)𝑓(𝜆^𝑠𝑑)≤𝐶𝑙𝑓(𝜆), for any fuzzy set 𝜆 in 𝑋.

Proof. (1)⇒(2) Let 𝑓 be fuzzy semicontinuous and 𝜆 a fuzzy set in 𝑋. Since Cl𝑓(𝜆) is fuzzy closed in 𝑌,𝑓⁻¹(Cl𝑓(𝜆)) is fuzzy semiclosed in 𝑋 such that 𝜆≤𝑓⁻¹(Cl𝑓(𝜆)). 𝜆≤𝑓⁻¹(𝑠Cl𝑓(𝜆)) gives 𝑠Cl𝜆≤𝑠Cl𝑓⁻¹(Cl𝑓(𝜆))=𝑓⁻¹(Cl𝑓(𝜆)). Therefore, 𝑓(𝜆^𝑠𝑑)≤𝑓(𝑠Cl𝜆)≤𝑓𝑓⁻¹(Cl𝑓(𝜆))≤Cl𝑓(𝜆). Consequently, 𝑓(𝜆^𝑠𝑑)≤Cl𝑓(𝜆).
(2)⇒(1) Suppose 𝑓(𝜆^𝑠𝑑)≤Cl𝑓(𝜆). Let 𝜇 be any fuzzy closed set in Y. We show that 𝑓⁻¹(𝜇) is fuzzy semiclosed in 𝑋. By our hypothesis, 𝑓([𝑓⁻¹(𝜇)]^𝑠𝑑)≤Cl𝑓(𝑓⁻¹(𝜇))≤Cl𝜇=𝜇, 𝑓([𝑓⁻¹(𝜇)]^𝑠𝑑)≤𝜇 gives [𝑓⁻¹(𝜇)]^𝑠𝑑≤𝑓⁻¹(𝑓[𝑓⁻¹(𝜇)]^𝑠𝑑)≤𝑓⁻¹(𝜇), or [𝑓⁻¹(𝜇)]^𝑠𝑑≤𝑓⁻¹(𝜇) implies that 𝑓⁻¹(𝜇) is fuzzy semiclosed in 𝑋. Thus, 𝑓 is fuzzy semicontinuous.

Finally, we characterize fuzzy irresolute functions via fuzzy semiderived set as follows.

Theorem 18. Let 𝑓∶𝑋→𝑌 be a function. Then, the following conditions are equivalent: (1)𝑓 is fuzzy irresolute,(2)𝑓(𝜆^𝑠𝑑)≤𝑠𝐶𝑙𝑓(𝜆), for any fuzzy set 𝜆 in 𝑋.

Acknowledgment

The authors are grateful to the referees whose valuable suggestions have improved the paper a lot.