Supra semi-compactness via supra topological spaces

ABSTRACT

In this paper, we utilize a supra semi-open sets notion to introduce and study the concepts of supra semi-compact (supra semi-Lindelöf) spaces, almost supra semi-compact (almost supra semi-Lindelöf) spaces and mildly supra semi-compact (mildly supra semi-Lindelöf) spaces in supra topological spaces. We investigate some properties of supra semi-closed and supra semi-clopen subsets of these spaces and we give the equivalent conditions for the concepts of supra semi-compact (supra semi-Lindelöf) spaces and mildly supra semi-compact (mildly supra semi-Lindelöf) spaces. With the help of examples, we illustrate the relationships among these concepts. We also derive some results which associate these spaces with some mappings.

KEYWORDS:

• Supra semi-open set
• supra semi-compactness
• almost (mildly) supra semi-compactness
• supra topological spaces

2010 AMS SUBJECT CLASSIFICATION:

• 54C08
• 54C10
• 54D30

1. Introduction

In 1963, Levine [1] introduced and studied a notion of semi-open sets in topological spaces. Mildly and almost compact spaces [2, 3] were introduced in 1974 and 1975, respectively. Dorsett [4] in 1980, presented a concept of semi-compact spaces and Mashhour et al. [5] in 1983, formulated a supra topological spaces concept and investigated some of its properties. In 2006, Min [6] introduced a notion of p-supracompactness by using the convergence of ultrapastakes and investigated some of its properties, and in 2013, Mustafa [7] introduced the concepts of supra b-compact and supra b-Lindelöf spaces. Al-shami [8] initiated some results concerning supra topologies and presented some types of supra compact spaces. He [9] formulated six new kinds of supra compact spaces by utilizing a supra α-open sets notion. In [10, 11], he investigated some concepts related to supra semi-open sets in supra topological spaces and supra topological ordered spaces, respectively.

The main purpose of this work is to introduce and study the concepts of supra semi-compact (supra semi-Lindelöf) spaces, almost supra semi-compact (almost supra semi-Lindelöf) spaces and mildly supra semi-compact (mildly supra semi-Lindelöf) spaces in supra topological spaces by using a notion of supra semi-open sets. The equivalent conditions for the concepts of supra semi-compact (supra semi-Lindelöf) spaces and mildly supra semi-compact (mildly supra semi-Lindelöf) spaces are given and investigated. Some results about the image of these spaces under some maps are discussed and some illustrative examples are supplied to show the relationships among these concepts.

2. Preliminaries

Here are some definitions and results required in the sequel.

Definition 2.1 ([5]):

A collection μ of subsets of is said to be a supra topology on X if X belongs to μ and the union of an arbitrary family of sets in μ belongs to μ.

A pair is called a supra topological space. Every member of μ is said to be supra open and its complement is said to be supra closed.

Remark 2.1:

Throughout this work, and denote to a topological space and a supra topological space, respectively, and the notations , , and stand for the set of natural numbers, the set of integer numbers, the set of rational numbers and the set of real numbers, respectively.

Definition 2.2 ([5]):

Let E be a subset of . Then the supra closure of E, denoted by , is the intersection of all supra closed sets containing E and the supra interior of E, denoted by , is the union of all supra open sets contained in E.

Definition 2.3 ([10]):

A subset E of is called supra semi-open if . The complement of a supra semi-open set is called supra semi-closed.

Definition 2.4 ([10]):

Let E be a subset of . Then the supra semi-closure of E, denoted by , is the intersection of all supra semi-closed sets containing E and the supra semi-interior of E, denoted by , is the union of all supra semi-open sets contained in E.

Definition 2.5 ([10]):

A map is said to be:

1. Supra semi-continuous (resp. Supra semi-irresolute) if the inverse image of each open (resp. supra semi-open) subset of Y is a supra semi-open subset of X.

2. Supra semi-open (resp. Supra semi-closed) if the image of each open (resp. closed) subset of X is a supra semi-open (resp. supra semi-closed) subset of Y.

Theorem 2.6 ([10]):

For a map we have the following two results:

1. g is supra semi-irresolute if and only if for each .

2. g is supra semi-closed if and only if for each .

Definition 2.7 ([8]):

A supra topological spaces is said to be:

1. Supra compact (resp. Supra Lindelöf) if every supra open cover of X has a finite (resp. countable) subcover.

2. Almost supra compact (resp. Almost supra Lindelöf) if every supra open cover of X has a finite (resp. countable) sub-collection, the supra closure of whose members cover X.

3. Mildly supra compact (resp. Mildly supra Lindelöf) if every supra clopen cover of X has a finite (resp. countable) subcover.

Definition 2.8:

A collection Λ of sets is said to have the finite intersection property (resp. countable intersection property) if every finite (resp. countable) sub-collection of Λ has a non-empty intersection.

3. Supra semi-compact spaces

In this section, we introduce the concepts of supra semi-compact and supra semi-Lindelöf spaces and study the equivalent conditions for them.

Definition 3.1:

A collection of supra semi-open subsets of is called a supra semi-open cover of a subset E of X provided that .

Definition 3.2:

A supra topological spaces is called supra semi-compact (resp. supra semi-Lindelöf) provided that every supra semi-open cover of X has a finite (resp. countable) subcover.

Now, we present two examples, the first one satisfies a concept of supra semi-compactness and the second one does not satisfy.

Example 3.3:

Let such that is finite be a supra topology on . In this supra topological space, we observe that a subset of is supra semi-open if it is supra open. Then is a supra semi-compact space.

Example 3.4:

Let or be a supra topology on . A collection forms a supra semi-open cover of . Since Λ has not a finite subcover of , then is not a supra semi-compact space.

The proofs of the following two propositions are straightforward and so will be omitted.

Proposition 3.5:

Every supra semi-compact space is supra semi-Lindelöf.

Proposition 3.6:

Every supra semi-compact resp. supra semi-Lindelöf space is supra compact resp. supra Lindelöf .

It can be seen from Example 3.4 that the converse of Proposition 3.5 fails. Also, If we replace by in Example 3.4, we obtain that is a supra compact space, whereas it is not supra semi-Lindelöf. So the converse of Proposition 3.6 fails as well.

Proposition 3.7:

Any finite resp. countable supra topological space is supra semi-compact resp. supra semi-Lindelöf .

Proof:

It is well known that the largest cover of any set X consists of the singleton subsets of X. So if X is finite (resp. countable), then the largest cover of X is finite (resp. countable). Hence the desired result is proved

Definition 3.8:

A subset E of is said to be supra semi-compact (resp. supra semi-Lindelöf) relative to X if every supra semi-open cover of E is reducible to a finite (resp. countable) subcover.

Proposition 3.9:

A finite resp. countable union of supra semi-compact resp. supra semi-Lindelöf subsets of is supra semi-compact resp. supra semi-Lindelöf .

Proof:

Straightforward.

Proposition 3.10:

Every supra semi-closed subset of a supra semi-compact resp. supra semi-Lindelöf space is supra semi-compact resp. supra semi-Lindelöf .

Proof:

Let be a supra semi-open cover of a supra semi-closed subset F of X. Then is a supra semi-open set and . Therefore . Since is supra semi-compact, then . So . Hence F is a supra semi-compact set.

The proof is similar in case of a supra semi-Lindelöf space.

It can be seen from Example 3.4 that is a supra semi-compact set. But it is not a supra semi-closed set. So the converse of Proposition 3.10 fails.

Theorem 3.11:

A supra topological space is supra semi-compact resp. supra semi-Lindelöf if and only if every collection of supra semi-closed subsets of X satisfies the finite resp. countable intersection property, has, itself, a non-empty intersection.

Proof:

We prove the theorem in case of supra semi-compactness and the case between parentheses made similarly.

Necessity: Let be a collection of supra semi-closed subsets of X which has the finite intersection property. Assume that . Then . Since X is supra semi-compact, then . Therefore . But this contradicts that Λ has the finite intersection property. Thus Λ has, itself, a non-empty intersection.

Sufficiency: Let be a supra semi-open cover of X. Suppose, to the contrary, that has no finite sub-cover. Then , for any . Now, . This implies that is a collection of supra semi-closed subsets of X which has the finite intersection property. Therefore . Thus . But this contradicts that is a cover of X. Hence is a supra semi-compact space.

Proposition 3.12:

If A is a supra semi-compact resp. supra semi-Lindelöf subset of X and B is a supra semi-closed subset of X, then is supra semi-compact resp. supra semi-Lindelöf .

Proof:

Let be a supra semi-open cover of . Then . Since A is supra semi-Lindelöf, then there exists a countable set S such that . Therefore . Thus is a supra semi-Lindelöf set.

Similarly, one can prove the proposition in case of a supra semi-compact space.

Theorem 3.13:

The supra semi-irresolute image of a supra semi-compact resp. supra semi-Lindelöf set is supra semi-compact resp. supra semi-Lindelöf .

Proof:

Let be a supra semi-irresolute map and let A be a supra semi-compact subset of X. Suppose that is a supra semi-open cover of . This automatically implies that . Since g is supra semi-irresolute, then is a supra semi-open set, for each . By hypotheses, A is supra semi-compact, then . So . Hence is supra semi-compact.

A similar proof can be given for the case between parentheses.

Theorem 3.14:

If is a bijective supra semi-open map and Y is a supra semi-compact resp. supra semi-Lindelöf space, then X is compact resp. Lindelöf .

Proof:

Let be an open cover of X. Then . Therefore . Since is a supra semi-open set, for each and Y is a supra semi-compact space, then . Since g is injective, then . Thus X is compact.

A similar proof can be given for the case between parentheses.

Corollary 3.15:

If is an injective supra semi-open map and is a supra semi-compact resp. supra semi-Lindelöf space, then X is compact resp. Lindelöf .

4. Almost supra semi-compact spaces

In this section, the concepts of almost supra semi-compact and almost supra semi-Lindelöf spaces are formulated and their properties are investigated with the help of examples.

Definition 4.1:

A supra topological spaces is called almost supra semi-compact (resp. almost supra semi-Lindelöf) provided that every supra semi-open cover of X has a finite (resp. countable) sub-collection, the supra semi-closure of whose members cover X.

In what follows, we give two examples, the first one satisfies a concept of almost supra semi-compactness and the second one does not satisfy.

Example 4.2:

Let be a supra topology on . In this supra topology, any supra open set is supra dense. We know that a set E is supra semi-open if and only if there exists a supra open set G such that . Then every supra semi open is supra dense as well. So is an almost supra semi-compact space. On the other hand, is a semi-open cover of . Since Λ has not a finite subcover, then is not supra semi-compact.

Example 4.3:

Let such that or be a supra topology on . Obviously, forms a supra semi-open cover of . Since , for all and , then is not an almost supra semi-compact space.

Proposition 4.4:

Every almost supra semi-compact space is almost supra semi-Lindelöf.

Proof:

The proof is obtained immediately from Definition 4.1.

In Example 3.4, we note that forms a supra semi-open cover of . Since Λ has not a finite sub-cover, the supra semi-closure of whose members cover X, then is not almost supra semi-compact. On the other hand, it is almost supra semi-Lindelöf. Hence the converse of the above proposition need not be true in general.

The proofs of the following two propositions are easy and so will be omitted.

Proposition 4.5:

A finite resp. countable union of almost supra semi-compact resp. almost supra semi-Lindelöf subsets of is almost supra semi-compact resp. almost supra semi-Lindelöf .

Proposition 4.6:

Every supra semi-compact resp. supra semi-Lindelöf space is almost supra semi-compact resp. almost supra semi-Lindelöf .

Corollary 4.7:

Any finite resp. countable supra topological space is almost supra semi-compact resp. almost supra semi-Lindelöf .

The converse of Proposition 4.6 is not always true as illustrated in the following example.

Example 4.8:

Let be a supra topology on . Obviously, is an almost supra semi-compact space but it is not supra semi-Lindelöf.

Definition 4.9:

A subset E of is said to be supra semi-clopen provided that it is supra semi-open and supra semi-closed.

Proposition 4.10:

Every supra semi-clopen subset of an almost supra semi-compact resp. almost supra semi-Lindelöf space is almost supra semi-compact resp. almost supra semi-Lindelöf .

Proof:

Let us prove the proposition in case of an almost supra semi-compact space and the other can be made similarly.

Let be a supra semi-open cover of a supra semi-clopen subset F of . Then is a supra semi-open set and . Therefore . Since is almost supra semi-compact, then . Thus . Hence F is an almost supra semi-compact set.

It can be seen from Example 4.3 that is an almost supra semi-compact set. But it is not a supra semi-clopen set. So the converse of Proposition 4.10 fails.

Proposition 4.11:

If A is an almost supra semi-compact resp. almost supra semi-Lindelöf subset of X and B is a supra semi-clopen subset of X, then is almost supra semi-compact resp. almost supra semi-Lindelöf .

Proof:

Let be a supra semi-open cover of . Then . Since A is almost supra semi-compact, then . Therefore . Thus is an almost supra semi-compact set.

A similar proof can be given for the case between parentheses.

Theorem 4.12:

The supra semi-irresolute image of an almost supra semi-compact resp. almost supra semi-Lindelöf set is almost supra semi-compact resp. almost supra semi-Lindelöf .

Proof:

Let be a supra semi-irresolute map and let A be an almost supra semi-compact subset of X. Suppose that be a supra semi-open cover of . This automatically implies that . Since g is supra semi-irresolute, then is a supra semi-open set, for each . By hypotheses, A is almost supra semi-compact, then . It follows, by Theorem 2.6, that , for each . So . Hence is an almost supra semi-compact set.

A similar proof can be given for the case between parentheses.

Corollary 4.13:

The supra semi-continuous image of an almost supra semi-compact resp. almost supra semi-Lindelöf set is almost compact resp. almost Lindelöf .

Theorem 4.14:

If is a bijective supra semi-open map and Y is almost supra semi-compact resp. almost supra semi-Lindelöf then X is almost compact resp. almost Lindelöf .

Proof:

Let be an open cover of X. Then . Therefore . Since is a supra semi-open set, for each and Y is almost supra semi-compact, then . Since g is bijective supra semi-open, then g is supra semi-closed. Therefore by Theorem 2.6, we obtain that . Thus . Hence X is almost compact.

A similar proof can be given for the case between parentheses.

Theorem 4.15:

If every collection of supra semi-closed subsets of satisfies the finite resp. countable intersection property, has, itself, a non-empty intersection, then is almost supra semi-compact resp. almost supra semi-Lindelöf .

Proof:

This is easily obtained from Theorem 3.11 and Proposition 4.6.

Assume that is the same as in Example 4.2. A collection consists of supra closed subsets of and has a finite intersection property. Whereas . So the converse of the above theorem is not always true.

5. Mildly supra semi-compact spaces

We present in this section the notions of mildly supra semi-compact and mildly supra semi-Lindelöf spaces. Also, we investigate the equivalent conditions for them and show the relationships between them with the help of examples.

Definition 5.1:

A supra topological space is called mildly supra semi-compact (resp. mildly supra semi-Lindelöf) provided that every supra semi-clopen cover of X has a finite (resp. countable) subcover.

It can be proved the following two propositions easily, so their proofs will be omitted.

Proposition 5.2:

Every mildly supra semi-compact resp. mildly supra semi-Lindelöf space is mildly supra compact resp. mildly supra Lindelöf .

Proposition 5.3:

Every mildly supra semi-compact space is mildly supra semi-Lindelöf.

The two examples below illustrate that the converse of the above propositions is not always true.

Example 5.4:

Let be a supra topological space, where μ is the usual topology. From the fact it is connected, we conclude that and are the only clopen subsets of . So it is mildly supra compact. On the other hand, a collection forms a supra semi-open cover of . Obviously, it is not a countable subcover, hence is not mildly supra semi-Lindelöf.

Example 5.5:

We replace by in Example 4.3. Then is mildly supra semi-Lindelöf. On the other hand, forms a supra semi-clopen cover of . Since Λ has not a finite subcover, then is not mildly supra semi-compact.

Proposition 5.6:

Every almost supra semi-compact resp. almost supra semi-Lindelöf space is mildly supra semi-compact resp. mildly supra semi-Lindelöf .

Proof:

Let be a supra semi-clopen cover of . Since is almost supra semi-compact, then . For each , we have , hence is mildly supra semi-compact.

A similar proof can be given for the case between parentheses.

Corollary 5.7:

Every supra semi-compact resp. supra semi-Lindelöf space is mildly supra semi-compact resp. mildly supra semi-Lindelöf .

The converse of the above corollary need not be true in general as the next example shows.

Example 5.8:

Let be a supra topology on . Since the only supra clopen subsets of are and , then is mildly supra semi-compact. On the other hand, a semi-open cover of has not a finite sub-cover. So is not supra semi-Lindelöf.

Proposition 5.9:

Every supra semi-clopen subset of a mildly supra semi-compact resp. mildly supra semi-Lindelöf space is mildly supra semi-compact resp. mildly supra semi-Lindelöf .

Proof:

Let be a supra semi-clopen cover of a supra semi-clopen subset F of X. Then is a supra semi-clopen set and . Therefore . Since X is mildly supra semi-compact, then . Thus . Hence F is a mildly supra semi-compact set.

The proof is similar in case of a mildly supra semi-Lindelöf space.

Proposition 5.10:

If A is a mildly supra semi-compact resp. mildly supra semi-Lindelöf subset of X and B is a supra semi-clopen subset of X, then is mildly supra semi-compact resp. mildly supra semi-Lindelöf .

Proof:

Let be a supra semi-clopen cover of . Then . Since A is mildly supra semi-compact, then . Therefore . Thus is a mildly supra semi-compact set.

The proof is similar in case of a mildly supra semi-Lindelöf space.

Theorem 5.11:

A supra topological space is mildly supra semi-compact resp. mildly supra semi-Lindelöf if and only if every collection of supra semi-clopen subsets of X, satisfies the finite resp. countable intersection property, has, itself, a non-empty intersection.

Proof:

We only prove the theorem when is mildly supra semi-compact, the other case can be made similarly.

Let be a collection of supra semi-clopen subsets of X which has the finite intersection property. Assume that . Then . Since X is mildly supra semi-compact, then . Therefore . But this contradicts that Λ has the finite intersection property. Thus Λ has, itself, a non-empty intersection.

Conversely, Let be a supra semi-clopen cover of X. Suppose, to the contrary, that has no finite sub-cover. Then , for any . Now, . This implies that is a collection of supra semi-clopen subsets of X which has the finite intersection property. Therefore. Thus . But this contradicts that is a cover of X. Hence is a mildly supra semi-compact space.

Proposition 5.12:

If is a bijective supra semi-open map and Y is mildly supra semi-compact, then X is mildly compact.

Proof:

Let be a clopen cover of X. Then . Therefore . Now, Y is mildly supra semi-compact, then . Since g is bijective supra semi-open, then . Hence X is mildly compact.

6. Conclusion

The concept of compactness is considered as one of fundamental concepts in topological spaces for its contribution in the study of many of topological problems. In this work, we present and study the concepts of supra semi-compact (supra semi-Lindelöf) spaces, almost (mildly) supra semi-compact spaces and almost (mildly) supra semi-Lindelöf spaces depending on a notion of supra semi-open sets. Also, we show the relationships among them with the help of examples. We give the equivalent conditions for the concepts of supra semi-compact (supra semi-Lindelöf) spaces and mildly supra semi-compact (mildly supra semi-Lindelöf) spaces. We verify that the semi-irresolute image of a supra semi-compact (an almost supra semi-compact) set is supra semi-compact (almost supra semi-compact). In an upcoming work, we plan to use a notion of somewhere dense sets [12] to study these concepts in topological spaces. In the end, we hope that the results obtained in this work will help research teams promote further study in supra topological spaces to carry out a general frame work for the practical applications.

Acknowledgments

The author express his sincere thanks to the reviewers for their valuable suggestions which helped to improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author.