More properties of generalized open sets in generalized topological spaces

Mohammad S. Sarsak

doi:10.1515/dema-2022-0027

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More properties of generalized open sets in generalized topological spaces

Mohammad S. Sarsak

Published/Copyright: August 13, 2022

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From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

Sarsak [M. S. Sarsak, On some properties of generalized open sets in generalized topological spaces, Demonstr. Math. 46 (2013), no. 2, 415–427] studied some properties of generalized open sets in generalized topological spaces (GTSs); the primary purpose of this article is to investigate more properties of generalized open sets in GTSs. We mainly study the behaviours of regular closed sets, semi-open sets, regular semi-open sets, preopen sets, and β -open sets in GTSs analogous to their behaviours in topological spaces.

Keywords: generalized topology; μ-space; μ-open; μ-regular closed; μ-semi-open; regular μ-semi-open; μ-preopen; μ-β-open

MSC 2010: 54A05; 54A10

1 Introduction and preliminaries

Several types of generalized open sets in topological spaces had been introduced and studied by several mathematicians; for instance, semi-open sets [1], preopen sets [2], and β -open sets [3].

Csàszàr [4] studied analogous generalized open sets in generalized topological spaces (GTSs). Sarsak [5] studied some properties of generalized open sets in GTSs; the primary purpose of this article is to investigate more properties of generalized open sets in GTSs. In Sections 2 and 3, we study the behaviours of μ -regular closed sets, μ -semi-open sets, μ -preopen sets, and μ - β -open sets in a subspace of a GTS; it is seen by examples that most of the behaviours of regular closed sets, semi-open sets, preopen sets, and β -open sets in topological spaces are invalid for GTSs. It is shown that if S is μ -semi-open (resp. μ -regular closed) and B is μ -preopen, then S ∩ B need not be μ B -semi-open (resp. μ B -regular closed), thus, answering a question posed in [6]. It is also shown that if S is μ -semi-open (resp. μ -preopen, μ - β -open) and B is μ -open, then S ∩ B need not be μ -semi-open (resp. μ -preopen, μ - β -open). We also show that if S is μ -semi-open and B is μ -preopen, then S ∩ B need not be μ S -preopen. The properties of being μ -semi-open, μ -preopen, and μ - β -open are, in general, not transitive in GTSs, that is if A is μ B -semi-open (resp. μ B -preopen, μ B - β -open), and B is μ -semi-open (resp. μ -preopen, μ - β -open), then A need not be μ -semi-open (resp. μ -preopen, μ - β -open). In a GTS, a μ B -regular closed subset of a μ -preopen set B need not be the intersection of a μ -regular closed set with B . In Section 4, a special interest is given to regular semi-open sets; it is shown that in a GTS, a regular μ B -semi-open subset of a μ -preopen set B need not be the intersection of a regular μ -semi-open set with B , and that the intersection of a regular μ -semi-open set S with a μ -preopen set B need not be regular μ B -semi-open.

A generalized topology (GT) [7] μ on a nonempty set X is a collection of subsets of X such that ∅ ∈ μ and μ is closed under arbitrary unions. For a GT μ on X , the pair ( X , μ ) will be called a GTS, elements of μ will be called μ -open sets, and a subset A of ( X , μ ) will be called μ -closed if X \ A is μ -open. A space X or ( X , μ ) will always mean a GTS. Clearly, a subset A of ( X , μ ) is μ -open if and only if for each x ∈ A , there exists U x ∈ μ such that x ∈ U x ⊂ A , or equivalently, A is the union of μ -open sets. A GT μ on X is called a strong GT [8] if X ∈ μ . A space ( X , μ ) is called a μ -space [9] if X ∈ μ , that is μ is a strong GT. A space ( X , μ ) is called a quasi-topological space [10] if μ is closed under finite intersections. Clearly, every topological space is a quasi-topological space, every quasi-topological space is a GTS, and a space ( X , μ ) is a topological space if and only if ( X , μ ) is both μ -space and quasi-topological space.

If A is a subset of a space ( X , μ ) , then the μ -closure of A [4], c μ ( A ) , is the intersection of all μ -closed sets containing A , and the μ -interior of A [4], i μ ( A ) , is the union of all μ -open sets contained in A . It was pointed out in [4] that each of the operators c μ and i μ are monotonic [11], i.e. if A ⊂ B ⊂ X , then c μ ( A ) ⊂ c μ ( B ) and i μ ( A ) ⊂ i μ ( B ) , idempotent [11], i.e. if A ⊂ X , then c μ ( c μ ( A ) ) = c μ ( A ) and i μ ( i μ ( A ) ) = i μ ( A ) , c μ is enlarging [11], i.e. if A ⊂ X , then c μ ( A ) ⊃ A , i μ is restricting [11], i.e. if A ⊂ X , then i μ ( A ) ⊂ A , and c μ ( A ) = X \ i μ ( X \ A ) .

Clearly, A is μ -closed if and only if A = c μ ( A ) , c μ ( A ) is the smallest μ -closed set containing A , and x ∈ c μ ( A ) if and only if any μ -open set containing x intersects A . Also, A is μ -open if and only if A = i μ ( A ) , i μ ( A ) is the largest μ -open set contained in A , and x ∈ i μ ( A ) if and only if there exists a μ -open set U such that x ∈ U ⊂ A .

If ( X , τ ) is a topological space and A ⊂ X , then A ¯ and Int A will stand, respectively, for the closure of A in X and the interior of A in X .

A subset A of a topological space ( X , τ ) is called regular open if A = Int A ¯ , or equivalently, if A = Int F , for some closed set F , and A is called regular closed if X \ A is regular open, or equivalently, if A = Int A ¯ , equivalently, A = U ¯ , for some open set U . A is called semi-open [1] (resp. preopen [2], β -open [3] or called semi-preopen [12]) if A ⊂ Int A ¯ (resp. A ⊂ Int A ¯ , A ⊂ Int A ¯ ¯ ). A is called semi-regular [13] if it is both semi-open and semi-closed, or equivalently, if there exists a regular open set U such that U ⊂ A ⊂ U ¯ ; in [14], the term regular semi-open was used for a semi-regular set. It is known that the arbitrary union of semi-open (resp. preopen, β -open) sets is semi-open (resp. preopen, β -open).

The families of semi-open (resp. preopen, β -open) subsets of a topological space ( X , τ ) will be denoted by S O ( X ) (resp. P O ( X ) , β ( X ) ). Clearly, if μ = S O ( X ) (resp. P O ( X ) , β ( X ) ), then ( X , μ ) is a μ -space. A is called semi-closed (resp. preclosed, β -closed) if X \ A is semi-open (resp. preopen, β -open). The semi-closure (resp. preclosure) of A denoted by scl ( A ) (resp. pcl ( A ) ) is the intersection of all semi-closed (resp. preclosed) sets containing A , or equivalently, the smallest semi-closed (resp. preclosed) set containing A . Clearly, A is semi-closed (resp. preclosed) if and only if A = scl ( A ) (resp. A = pcl ( A ) ). Andrijević [12] pointed out that scl ( A ) = A ∪ Int A ¯ and that pcl ( A ) = A ∪ Int A ¯ .

For the concepts and terminology not defined here, the reader is referred to [15]. In concluding this section, we recall the following definitions and facts for their importance in the material of our article.

Definition 1.1

[4] Let A be a subset of a space ( X , μ ) . Then A is called

μ -semi-open if A ⊂ c μ ( i μ ( A ) ) ;
μ -preopen if A ⊂ i μ ( c μ ( A ) ) ;
μ - β -open if A ⊂ c μ ( i μ ( c μ ( A ) ) ) .

As in [4], for a space ( X , μ ) , we will denote the class of μ -semi-open sets by σ ( μ ) or σ , the class of μ -preopen sets by π ( μ ) or π , and the class of μ - β -open sets by β ( μ ) or β .

Proposition 1.2

[4] Let ( X , μ ) be a space. Then each of the families σ , π , and β is a GT.

Proposition 1.3

[4] Let ( X , μ ) be a space. Then

μ ⊂ σ ⊂ β ;
μ ⊂ π ⊂ β .

Definition 1.4

[16] Let A be a subset of a space ( X , μ ) . Then A is called

μ -regular closed if A = c μ ( i μ ( A ) ) ;
μ -regular open if X \ A is μ -regular closed, or equivalently, if A = i μ ( c μ ( A ) ) ;
μ -semi-closed if X \ A is μ -semi-open;
μ -preclosed if X \ A is μ -preopen;
μ - β -closed if X \ A is μ - β -open.

Definition 1.5

[5] Let A be a subset of a space ( X , μ ) . Then A is called

μ -dense if c μ ( A ) = X ;
μ -codense if X \ A is μ -dense, or equivalently, if i μ ( A ) = ∅ ;
regular μ -semi-open if there exists a μ -regular open set U such that U ⊂ A ⊂ c μ ( U ) , or equivalently, if A is both μ -semi-open and μ -semi-closed.

Remark 1.6

[5] Let A be a subset of a space ( X , μ ) . If A is μ -regular open ( μ -regular closed), then A is regular μ -semi-open.

Remark 1.7

[5] Let A be a subset of a μ -space ( X , μ ) . If A is μ -dense, then A is μ -preopen.

2 Subspaces

The primary purpose of this section is to study the behaviours of μ -semi-open sets, μ -dense sets, μ -preopen sets, and μ - β -open sets in a subspace of a GTS.

Definition 2.1

[17] Let A be a nonempty subset of a space ( X , μ ) . The generalized subspace topology on A is the collection { U ∩ A : U ∈ μ } , which will be denoted by μ A . The generalized subspace A is the GTS ( A , μ A ) .

Remark 2.2

[17] Let A be a nonempty subset of a space ( X , μ ) . Then it is easy to see that

If ( X , μ ) is a μ -space, then ( A , μ A ) is a μ A -space;
A subset B of A is μ A -closed if and only if B = F ∩ A , for some μ -closed set F .

Proposition 2.3

Let ( X , μ ) be a space and A ⊂ B ⊂ X . Then

[6] If A is μ -open (resp. μ -closed), then A is μ B -open (resp. μ B -closed);
If A is μ B -closed and B is μ -closed, then A is μ -closed.

Proof

Follows from the definition of the generalized subspace topology and from Remark 2.2(ii) since A = A ∩ B .
By Remark 2.2(ii), A = F ∩ B , for some μ -closed set F , but B is μ -closed and any intersection of μ -closed sets is μ -closed, so A is μ -closed.□

We recall the following facts.

Proposition 2.4

Let A ⊂ B ⊂ X , where X is a topological space. If B is preopen, then

Int B A ¯ B = Int A ¯ ∩ B .
scl B ( A ) = scl ( A ) ∩ B .

Corollary 2.5

(Proposition 1.2 of [18]) Let ( X , τ ) be a topological space, and A ⊂ B ⊂ X , where B is preopen in X . Then A is semi-open (resp. semi-closed) in B if and only if A = S ∩ B , for some semi-open (resp. semi-closed) set S in X .

Proof

Follows from Proposition 2.4(ii).□

Corollary 2.6

Let ( X , τ ) be a topological space, and A ⊂ B ⊂ X , where B is preopen in X . If A is semi-closed in X , then A is semi-closed in B .

Proof

Follows from Corollary 2.5 since A = A ∩ B .□

The next proposition was shown in [6]; it shows that the condition “preopen” is not essential for the necessity of Corollary 2.5. We will, however, state its proof for the convenience of the reader.

Lemma 2.7

(Proposition 3.8 of [5]) Let ( X , μ ) be a space, and A ⊂ X . Then A is μ -semi-open if and only if U ⊂ A ⊂ c μ ( U ) , for some μ -open set U .

Lemma 2.8

(Lemma 3.11 of [16]) Let ( X , μ ) be a space, and A ⊂ B ⊂ X . Then c μ B ( A ) = c μ ( A ) ∩ B .

Proposition 2.9

[6] Let ( X , μ ) be a space, and A ⊂ B ⊂ X . Then

If A is μ B -semi-open (resp. μ B -semi-closed), then A = S ∩ B , for some μ -semi-open (resp. μ -semi-closed) set S ;
If A is μ -semi-open, then A is μ B -semi-open.

Proof

(i) Let A be μ B -semi-open. Then it follows from Lemmas 2.7 and 2.8 that there exists a μ -open set U such that

U ∩ B ⊂ A ⊂ c μ ( U ∩ B ) ∩ B ⊂ c μ ( U ) .

Let S = U ∪ A . Since A ⊂ c μ ( U ) , then clearly,

U ⊂ S ⊂ c μ ( U ) .

Thus by Lemma 2.7, S is μ -semi-open. Since U ∩ B ⊂ A ,

S ∩ B = ( U ∩ B ) ∪ A = A .

Now if A be μ B -semi-closed, then B \ A is μ B -semi-open, and thus, B \ A = S ∩ B , for some μ -semi-open set S . Hence, A = ( X \ S ) ∩ B , and X \ S is μ -semi-closed.

(ii) Since A is μ -semi-open, it follows from Lemma 2.7 that there exists a μ -open U such that

U ⊂ A ⊂ c μ ( U ) .

But A ⊂ B , so

U ⊂ A ⊂ c μ ( U ) ∩ B .

Since U is μ -open, and U ⊂ B , so by Proposition 2.3(i), U is μ B -open. Also by Lemma 2.8, c μ ( U ) ∩ B = c μ B ( U ) . Thus by Lemma 2.7, A is μ B -semi-open.□

We point out here that, in the particular case of a topological space, Proposition 2.9(ii) reduces to Theorem 6 of [1].

Remark 2.10

[6] From Proposition 2.9(i), we observe that the condition “preopen” is not essential for the necessity of Corollary 2.5.

Corollary 2.11

Let ( X , μ ) be a space, and A ⊂ B ⊂ X . If A is μ B -semi-closed and B is μ -semi-closed, then A is μ -semi-closed.

Proof

Follows from Proposition 2.9(i) since any intersection of μ -semi-closed sets is μ -semi-closed.□

Proposition 2.12

Let ( X , μ ) be a space, and A ⊂ B ⊂ X , where A is μ -preopen. Then A is μ B -preopen.

Proof

Since A is μ -preopen, A ⊂ i μ ( c μ ( A ) ) . Let x ∈ A . Then there exists a μ -open set U such that x ∈ U ⊂ c μ ( A ) . Thus, x ∈ U ∩ B ⊂ c μ ( A ) ∩ B . By Lemma 2.8, c μ ( A ) ∩ B = c μ B ( A ) . Therefore, x ∈ i μ B ( c μ B ( A ) ) . Thus, A ⊂ i μ B ( c μ B ( A ) ) . Hence, A is μ B -preopen.□

We point out here that, in the particular case of a topological space, Proposition 2.12 reduces to the fact that if X is a topological space, and A ⊂ B ⊂ X , where A is preopen, then A is preopen in B .

Proposition 2.13

Let ( X , μ ) be a space, and A ⊂ B ⊂ X . Then

If A is μ B -dense (resp. μ B -codense), then A = S ∩ B , for some μ -dense (resp. μ -codense) set S ;
If A is μ -dense, then A is μ B -dense;
If A is μ B -dense, B is μ -dense, then A is μ -dense.

Proof

Since A is μ B -dense, by Lemma 2.8, c μ B ( A ) = c μ ( A ) ∩ B = B . Thus, B ⊂ c μ ( A ) . Let S = A ∪ ( X \ B ) . Then clearly S is μ -dense, and A = S ∩ B . Now if A be μ B -codense, then B \ A is μ B -dense, and thus, B \ A = S ∩ B , for some μ -dense set S . Hence, A = ( X \ S ) ∩ B , and X \ S is μ -codense.
Since A is μ -dense, c μ ( A ) = X . Thus by Lemma 2.8, c μ B ( A ) = c μ ( A ) ∩ B = B . Hence, A is μ B -dense.
Since A is μ B -dense, B ⊂ c μ ( A ) , but c μ ( A ) is μ -closed, so c μ ( B ) ⊂ c μ ( A ) . Since B is μ -dense, c μ ( B ) = X . Thus, c μ ( A ) = X , that is, A is μ -dense.□

We recall the following facts.

Proposition 2.14

Let A ⊂ B ⊂ X , where X is a topological space. If B is semi-open, then

Int B A ¯ B = Int A ¯ ∩ B .
pcl B ( A ) = pcl ( A ) ∩ B .

Corollary 2.15

Let ( X , τ ) be a topological space, and A ⊂ B ⊂ X , where B is semi-open in X . Then A is preopen (resp. preclosed) in B if and only if A = P ∩ B , for some preopen (resp. preclosed) set P in X .

Proof

Follows from Proposition 2.14(ii).□

The next proposition shows that the condition “semi-open” is not essential for the necessity of Corollary 2.15.

Lemma 2.16

[19] A subset A of a topological space X is preopen if and only if A = U ∩ D , where U is open and D is dense.

Proposition 2.17

Let A ⊂ B ⊂ X , where X is a topological space. If A is preopen (resp. preclosed) in B , then A = P ∩ B , for some preopen (resp. preclosed) set P in X .

Proof

The case when A is preopen in B is Lemma 2.1 of [20]; however, the proof is a direct consequence of Proposition 2.13(i) and Lemma 2.16. Assume now that A is preclosed in B . Then B \ A is preopen in B , and therefore, B \ A = P ∩ B , for some preopen set P in X . Thus, A = ( X \ P ) ∩ B , and X \ P is preclosed in X .□

Lemma 2.18

[21] A subset A of a topological space X is β -open if and only if A = S ∩ D , where S is semi-open and D is dense.

Corollary 2.19

Let A ⊂ B ⊂ X , where X is a topological space. If A is β -open (resp. β -closed) in B , then A = K ∩ B , for some β -open (resp. β -closed) set K in X .

Proof

If A is β -open in B , then the result follows immediately from Proposition 2.9(i), Proposition 2.13(i), and Lemma 2.18. Assume that A is β -closed in B . Then B \ A is β -open in B , and therefore, B \ A = K ∩ B , for some β -open set K in X . Thus, A = ( X \ K ) ∩ B , and X \ K is β -closed in X .□

Question 1. Are the results of Proposition 2.17 and Corollary 2.19 still true for GTSs?

3 Counter examples

The primary purpose of this section is to show that most of the behaviours of regular closed sets, semi-open sets, preopen sets, and β -open sets in topological spaces are, in general, invalid for GTSs. We recall the following facts.

Proposition 3.1

Let ( X , τ ) be a topological space. Then

[22] If S is semi-open and B is preopen, then S ∩ B is semi-open in B.
[23] If B is semi-open, S is open, then S ∩ B is semi-open.
[22] If S is semi-open and B is preopen, then S ∩ B is preopen in S.
[2] If B is preopen, S is open, then S ∩ B is preopen.
[22] Let A ⊂ B ⊂ X . If A is preopen in B , and B is preopen in X , then A is preopen in X .
[23] Let A ⊂ B ⊂ X . If A is semi-open in B , and B is semi-open in X , then A is semi-open in X .
[3] Let A ⊂ B ⊂ X . If A is β -open in B, and B is β -open in X , then A is β -open in X .
[12] If B is β -open, S is open, then S ∩ B is β -open.
[24] Let A ⊂ B ⊂ X , where B is preopen in X . Then A is regular closed in B if and only if A = P ∩ B , for some regular closed set P in X .

We will show by examples that all parts of Proposition 3.1 need not be true, in general, for GTSs.

The following question was posed in [6].

Question 2. [6] Let ( X , μ ) be a GTS, where S is μ -semi-open and B is μ -preopen. Is S ∩ B necessarily μ B -semi-open?

The following two examples show that Proposition 3.1(i) and (ii) need not be true, in general, for GTSs. Thus, answering Question 2.

Example 3.2

Let

μ = { ∅ , { 1 , 2 , 3 } , { 2 , 3 , 4 , 5 } , { 1 , 4 , 5 } , { 1 , 2 , 3 , 4 , 5 } } ,

and let X = { 1 , 2 , 3 , 4 , 5 , 6 } . Then μ is a GT on X , and ( X , μ ) is not a μ -space.

If S = { 1 , 2 , 3 , 4 } , then S is μ -semi-open as

S ⊂ c μ ( i μ ( S ) ) = c μ ( { 1 , 2 , 3 } ) = X .

Let B = { 1 , 4 , 5 } . Then μ B = { B , ∅ , { 1 } , { 4 , 5 } } , and S ∩ B = { 1 , 4 } is not μ B -semi-open as

{ 1 , 4 } ⊄ c μ B ( i μ B ( { 1 , 4 } ) ) = c μ B ( { 1 } ) = { 1 } .

Thus by Proposition 2.9(ii), S ∩ B is not μ -semi-open. Note also that

{ 1 , 4 } ⊄ c μ ( i μ ( { 1 , 4 } ) ) = c μ ( ∅ ) = { 6 } .

Example 3.3

Let μ = { X , ∅ , { 1 , 2 , 3 } , { 2 , 3 , 4 , 5 } , { 1 , 4 , 5 } } , where X = { 1 , 2 , 3 , 4 , 5 } . Then ( X , μ ) is a μ -space. If S = { 1 , 2 , 3 , 4 } , then S is μ -semi-open as

S ⊂ c μ ( i μ ( S ) ) = c μ ( { 1 , 2 , 3 } ) = X .

Let B = { 1 , 4 , 5 } . Then μ B = { B , ∅ , { 1 } , { 4 , 5 } } , and S ∩ B = { 1 , 4 } is not μ B -semi-open as

{ 1 , 4 } ⊄ c μ B ( i μ B ( { 1 , 4 } ) ) = c μ B ( { 1 } ) = { 1 } .

Thus by Proposition 2.9(ii), S ∩ B is not μ -semi-open. Note also that

{ 1 , 4 } ⊄ c μ ( i μ ( { 1 , 4 } ) ) = c μ ( ∅ ) = ∅ .

The next two examples show that Proposition 3.1(iii), (iv), and (viii) need not be true, in general, for GTSs.

Example 3.4

Let μ = { ∅ , { 1 , 2 , 3 } , { 3 , 4 , 5 } , { 1 , 2 , 3 , 4 , 5 } } , and let X = { 1 , 2 , 3 , 4 , 5 , 6 } . Then μ is a GT on X , and ( X , μ ) is not a μ -space. Let

S = { 1 , 2 , 3 } and B = { 1 , 2 , 4 , 5 } .

Then μ S = { S , ∅ , { 3 } } . Also, B is μ -preopen because

{ 1 , 2 , 4 , 5 } ⊂ i μ ( c μ ( { 1 , 2 , 4 , 5 } ) ) = i μ ( X ) = { 1 , 2 , 3 , 4 , 5 } .

However, S ∩ B = { 1 , 2 } is not μ S -preopen because

{ 1 , 2 } ⊄ i μ S ( c μ S ( { 1 , 2 } ) ) = i μ S ( { 1 , 2 } ) = ∅ .

Thus, by Proposition 2.12, S ∩ B is not μ -preopen. Note also that

{ 1 , 2 } ⊄ i μ ( c μ ( { 1 , 2 } ) ) = i μ ( { 1 , 2 , 6 } ) = ∅ .

Observe also that B is μ - β -open as it is μ -preopen, and that S ∩ B is not even μ - β -open as

{ 1 , 2 } ⊄ c μ ( i μ ( c μ ( { 1 , 2 } ) ) ) = c μ ( ∅ ) = { 6 } .

Example 3.5

Let μ = { X , ∅ , { 2 , 3 , 4 } , { 1 , 3 , 5 } } , where X = { 1 , 2 , 3 , 4 , 5 } . Then ( X , μ ) is a μ -space. Let S = { 2 , 3 , 4 } and B = { 4 , 5 } . Then B is μ -preopen by Remark 1.7 as it is μ -dense, also, μ S = { S , ∅ , { 3 } } . Now S ∩ B = { 4 } is not μ S -preopen as

{ 4 } ⊄ i μ S ( c μ S ( { 4 } ) ) = i μ S ( { 2 , 4 } ) = ∅ .

Thus by Proposition 2.12, S ∩ B is not μ -preopen. Note also that

{ 4 } ⊄ i μ ( c μ ( { 4 } ) ) = i μ ( { 2 , 4 } ) = ∅ .

Observe also that B is μ - β -open as it is μ -preopen, and that S ∩ B is not even μ - β -open as

{ 4 } ⊄ c μ ( i μ ( c μ ( { 4 } ) ) ) = c μ ( ∅ ) = ∅ .

Before proceeding on the study of transitivity properties of generalized open sets in GTSs, we would like to point out first that the property of being open is not transitive in GTSs as the following two easy examples tell.

Example 3.6

Let μ = { ∅ , { 1 , 2 } , { 2 , 3 } , { 1 , 2 , 3 } } , where X = { 1 , 2 , 3 , 4 } . Then μ is a GT on X , and ( X , μ ) is not a μ -space. Let B = { 1 , 2 } . Then μ B = { B , ∅ , { 2 } } . Let A = { 2 } . Then A is μ B -open, and B is μ -open, but A is not μ -open.

Example 3.7

Let μ = { X , ∅ , { 1 , 2 } , { 2 , 3 } } , where X = { 1 , 2 , 3 } . Then ( X , μ ) is a μ -space. Let B = { 1 , 2 } . Then μ B = { B , ∅ , { 2 } } . Let A = { 2 } . Then A is μ B -open, and B is μ -open, but A is not μ -open.

The next two examples show that Proposition 3.1(v) and (vii) need not be true, in general, for GTSs.

Example 3.8

Let ( X , μ ) be the space of Example 3.4, that is μ = { ∅ , { 1 , 2 , 3 } , { 3 , 4 , 5 } , { 1 , 2 , 3 , 4 , 5 } } , where

X = { 1 , 2 , 3 , 4 , 5 , 6 } .

If B = { 1 , 2 , 4 , 5 } , then B is μ -preopen as seen before. Thus, B is μ - β -open. Now μ B = { B , ∅ , { 1 , 2 } , { 4 , 5 } } . Let A = { 1 , 2 } . Then A is μ B -preopen as it is μ B -open. Thus, A is μ B - β -open. However, A is not μ - β -open as

{ 1 , 2 } ⊄ c μ ( i μ ( c μ ( { 1 , 2 } ) ) ) = c μ ( i μ ( { 1 , 2 , 6 } ) ) = c μ ( ∅ ) = { 6 } .

Thus, A is not μ -preopen as was seen also in Example 3.4.

Example 3.9

Let ( X , μ ) be the space of Example 3.5, that is μ = { X , ∅ , { 2 , 3 , 4 } , { 1 , 3 , 5 } } , where X = { 1 , 2 , 3 , 4 , 5 } . If B = { 4 , 5 } , then B is μ -preopen as seen before. Thus, B is μ - β -open. Now μ B = { B , ∅ , { 4 } , { 5 } } . Let A = { 4 } . Then A is μ B -preopen as it is μ B -open. Thus, A is μ B - β -open. However, A is not μ - β -open as

{ 4 } ⊄ c μ ( i μ ( c μ ( { 4 } ) ) ) = c μ ( i μ ( { 2 , 4 } ) ) = c μ ( ∅ ) = ∅ .

Thus, A is not μ -preopen as was seen also in Example 3.5.

The next two examples show that Proposition 3.1(vi) need not be true, in general, for GTSs.

Example 3.10

Let ( X , μ ) be the space of Example 3.2, that is μ = { ∅ , { 1 , 2 , 3 } , { 2 , 3 , 4 , 5 } , { 1 , 4 , 5 } , { 1 , 2 , 3 , 4 , 5 } } , where X = { 1 , 2 , 3 , 4 , 5 , 6 } . If B = { 1 , 2 , 3 , 4 } , then B is μ -semi-open as seen before. Now

μ B = { B , ∅ , { 1 , 2 , 3 } , { 2 , 3 , 4 } , { 1 , 4 } } .

Let A = { 1 , 3 , 4 } . Then A is μ B -semi-open as

A ⊂ c μ B ( i μ B ( A ) ) = c μ B ( { 1 , 4 } ) = B .

However, A is not μ -semi-open as

A ⊄ c μ ( i μ ( A ) ) = c μ ( ∅ ) = { 6 } .

Example 3.11

Let ( X , μ ) be the space of Example 3.3, that is μ = { X , ∅ , { 1 , 2 , 3 } , { 2 , 3 , 4 , 5 } , { 1 , 4 , 5 } } , where X = { 1 , 2 , 3 , 4 , 5 } . If B = { 1 , 2 , 3 , 4 } , then B is μ -semi-open as seen before. Now

μ B = { B , ∅ , { 1 , 2 , 3 } , { 2 , 3 , 4 } , { 1 , 4 } } .

Let A = { 1 , 3 , 4 } . Then A is μ B -semi-open as

A ⊂ c μ B ( i μ B ( A ) ) = c μ B ( { 1 , 4 } ) = B .

However, A is not μ -semi-open as

A ⊄ c μ ( i μ ( A ) ) = c μ ( ∅ ) = ∅ .

The following two examples show that the sufficiency of Proposition 3.1(ix) need not be true, in general, for GTSs.

Example 3.12

Let ℬ = { { 1 , 2 , 3 } , { 3 , 4 , 5 } , { 5 , 6 , 7 } } , and let μ = { ∅ , all possible unions of members of ℬ } . Then μ is a GT on X , where X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } , and ( X , μ ) is not a μ -space. Now let B = { 3 , 4 , 6 , 7 } . Then B is μ -preopen because

{ 3 , 4 , 6 , 7 } ⊂ i μ ( c μ ( { 3 , 4 , 6 , 7 } ) ) = i μ ( X ) = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } .

Let P = { 4 , 5 , 6 , 7 , 8 } . Then P is μ -regular closed because

c μ ( i μ ( P ) ) = c μ ( { 5 , 6 , 7 } ) = P .

Observe that μ B = { B , ∅ , { 3 } , { 3 , 4 } , { 6 , 7 } , { 3 , 6 , 7 } } . Now, P ∩ B = { 4 , 6 , 7 } is not μ B -regular closed because

c μ B ( i μ B ( { 4 , 6 , 7 } ) ) = c μ B ( { 6 , 7 } ) = { 6 , 7 } ≠ { 4 , 6 , 7 } .

Example 3.13

Let ℬ = { { 1 , 2 , 3 } , { 3 , 4 , 5 } , { 5 , 6 , 7 } } , and let μ = { ∅ , all possible unions of members of ℬ } . Then ( X , μ ) is a μ -space, where X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . Now let B = { 3 , 4 , 6 , 7 } . Then B is μ -preopen by Remark 1.7 as it is μ -dense. Let P = { 4 , 5 , 6 , 7 } . Then P is μ -regular closed because

c μ ( i μ ( P ) ) = c μ ( { 5 , 6 , 7 } ) = P .

Observe that μ B = { B , ∅ , { 3 } , { 3 , 4 } , { 6 , 7 } , { 3 , 6 , 7 } } . Now, P ∩ B = { 4 , 6 , 7 } is not μ B -regular closed because

c μ B ( i μ B ( { 4 , 6 , 7 } ) ) = c μ B ( { 6 , 7 } ) = { 6 , 7 } ≠ { 4 , 6 , 7 } .

The following two examples show that the necessity of Proposition 3.1(ix) need not be true, in general, for GTSs.

Example 3.14

Let ( X , μ ) be the space of Example 3.4, that is μ = { ∅ , { 1 , 2 , 3 } , { 3 , 4 , 5 } , { 1 , 2 , 3 , 4 , 5 } } , and X = { 1 , 2 , 3 , 4 , 5 , 6 } . Then μ is a GT on X , and ( X , μ ) is not a μ -space. If B = { 2 , 5 } , then B is μ -preopen because

{ 2 , 5 } ⊂ i μ ( c μ ( { 2 , 5 } ) ) = i μ ( X ) = { 1 , 2 , 3 , 4 , 5 } .

Now μ B = { B , ∅ , { 2 } , { 5 } } . Let A = { 2 } . Then A is clearly μ B -regular closed as it is both μ B -open and μ B -closed. We will see that there is no μ -regular closed set P such that A = P ∩ B . Observe that the μ -closed sets are X , { 4 , 5 , 6 } , { 1 , 2 , 6 } , and { 6 } . Since i μ ( { 4 , 5 , 6 } ) = i μ ( { 1 , 2 , 6 } ) = i μ ( { 6 } ) = ∅ , and c μ ( ∅ ) = { 6 } , it follows that { 4 , 5 , 6 } and { 1 , 2 , 6 } are not μ -regular closed. Thus, the only μ -regular closed sets are X and {6}. Hence, there is no μ -regular closed set P such that A = P ∩ B .

Example 3.15

Let ( X , μ ) be the space of Example 3.5, that is μ = { X , ∅ , { 2 , 3 , 4 } , { 1 , 3 , 5 } } , where X = { 1 , 2 , 3 , 4 , 5 } . If B = { 4 , 5 } , then B is μ -preopen as seen before. Now μ B = { B , ∅ , { 4 } , { 5 } } . Let A = { 4 } . Then A is clearly μ B -regular closed as it is both μ B -open and μ B -closed. We will see that there is no μ -regular closed set P such that A = P ∩ B . Observe that the μ -closed sets are X , ∅ , { 1 , 5 } , and { 2 , 4 } . Since i μ ( { 1 , 5 } ) = i μ ( { 2 , 4 } ) = ∅ , and c μ ( ∅ ) = ∅ , it follows that { 1 , 5 } and { 2 , 4 } are not μ -regular closed. Thus, the only μ -regular closed sets are X and ∅ . Hence, there is no μ -regular closed set P such that A = P ∩ B .

4 Regular μ -semi-open sets

This section is mainly devoted to discussing the behaviour of regular semi-open sets in a preopen subspace of a GTS.

The following (probably) known result is an equivalent form of Proposition 3.1(ix); however, we will state its direct proof.

Lemma 4.1

Let ( X , τ ) be a topological space. If A ⊂ B ⊂ X , where B is preopen in X , then A is regular open in B if and only if A = U ∩ B , for some regular open set U in X .

Proof

A is regular open in B ⇔ B A is regular closed in B ⇔ B \ A = P ∩ B for some regular closed set P in X (by Proposition 3.1(ix)) ⇔ A = ( X \ P ) ∩ B , and X \ P is regular open in X . □

Corollary 4.2

Let ( X , τ ) be a topological space, and A ⊂ B ⊂ X , where B is preopen in X . If A is regular open (resp. regular closed) in X , then A is regular open (resp. regular closed) in B .

Proof

Follows from Proposition 3.1(ix) and Lemma 4.1 since A = A ∩ B .□

We now recall the following corollary describing the behaviour of regular semi-open sets in a preopen subspace of a topological space, we state its proof for the convenience of the reader.

Corollary 4.3

Let ( X , τ ) be a topological space. If A ⊂ B ⊂ X , where B is preopen in X , then

[25] A is regular semi-open in B if and only if A = S ∩ B , for some regular semi-open set S in X ;
[26] If A is regular semi-open in X , then A is regular semi-open in B .

Proof

(i) Necessity. Let A be regular semi-open in B . Then by Lemma 4.1, there exists a regular open set U in X such that

U ∩ B ⊂ A ⊂ U ∩ B ¯ ∩ B ⊂ U ¯ .

Let S = U ∪ A . Since A ⊂ U ¯ ,

U ⊂ S ⊂ U ¯ .

Thus, S is regular semi-open in X . Since U ∩ B ⊂ A ,

S ∩ B = ( U ∩ B ) ∪ A = A .

Sufficiency. Let A = S ∩ B , for some regular semi-open set S in X . Then S is both semi-open and semi-closed in X . From Corollary 2.5, it follows that A is both semi-open and semi-closed in B . Hence, A is regular semi-open in B .

(ii) Follows from Corollary 4.3(i) since A = A ∩ B .□

The following example shows that the sufficiency of Corollary 4.3(i) need not be true, in general, for GTSs.

Example 4.4

Let ( X , μ ) be the space of Example 3.13, that is μ = { ∅ , all possible unions of members of ℬ } , where

ℬ = { { 1 , 2 , 3 } , { 3 , 4 , 5 } , { 5 , 6 , 7 } } ,

and X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . Then by Remark 1.6, P = { 4 , 5 , 6 , 7 } is regular μ -semi-open as it is μ -regular closed. Let B = { 3 , 4 , 6 , 7 } . Then μ B = { B , ∅ , { 3 } , { 3 , 4 } , { 6 , 7 } , { 3 , 6 , 7 } } , and B is μ -preopen. However, P ∩ B = { 4 , 6 , 7 } is not regular μ B -semi-open as it is not μ B -semi-open. Observe that

{ 4 , 6 , 7 } ⊄ c μ B ( i μ B ( { 4 , 6 , 7 } ) ) = c μ B ( { 6 , 7 } ) = { 6 , 7 } .

The following example shows that the necessity of Corollary 4.3(i) need not be true, in general, for GTSs.

Example 4.5

Let ( X , μ ) be the space of Example 3.15, that is μ = { X , ∅ , { 2 , 3 , 4 } , { 1 , 3 , 5 } } , where X = { 1 , 2 , 3 , 4 , 5 } . Then B = { 4 , 5 } is μ -preopen as seen before, and μ B = { B , ∅ , { 4 } , { 5 } } . Let A = { 4 } . Then by Remark 1.6, A is regular μ B -semi-open as it is μ B -regular closed. Since the only μ -regular closed sets are X and ∅ , the only μ -regular open sets are X and ∅ . Thus, the only regular μ -semi-open sets are X and ∅ . Hence, there is no regular μ -semi-open set S such that A = S ∩ B .

At the end of this section, we recall the following corollary telling that the property of regular semi-open is transitive in topological spaces.

Corollary 4.6

[26] Let ( X , τ ) be a topological space, and A ⊂ B ⊂ X . If A is regular semi-open in B , and B is regular semi-open in X , then A is regular semi-open in X .

Proof

Follows from Corollary 2.11 and Proposition 3.1(vi).□

Question 3. Is the result of Corollary 4.6 still true for GTSs?

Acknowledgements

The author would like to thank the referees for their valuable suggestions and comments.

Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author states no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2021-08-11

Revised: 2022-02-16

Accepted: 2022-05-30

Published Online: 2022-08-13

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0027

Keywords for this article

generalized topology; μ-space; μ-open; μ-regular closed; μ-semi-open; regular μ-semi-open; μ-preopen; μ-β-open

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