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.


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 bcompact 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 semicompact (mildly supra semi-Lindelöf) spaces in supra topological spaces by using a notion of supra semiopen 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.

Preliminaries
Here are some definitions and results required in the sequel.
Definition 2.1 ( [5]): A collection μ of subsets of 2 X 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 (X, m) 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, (X, t) and (X, m) denote to a topological space and a supra topological space, respectively, and the notations N, Z, Q and R 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 (X, m). Then the supra closure of E, denoted by cl m (E), is the intersection of all supra closed sets containing E and the supra interior of E, denoted by int m (E), is the union of all supra open sets contained in E.
Definition 2.7 ( [8]): A supra topological spaces (X, m) is said to be: (i) Supra compact (resp. Supra Lindelöf) if every supra open cover of X has a finite (resp. countable) subcover. (ii) 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. (iii) 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) subcollection of Λ has a non-empty intersection.

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.  The proofs of the following two propositions are straightforward and so will be omitted.
It can be seen from Example 3.4 that the converse of Proposition 3.5 fails. Also, If we replace Q by R in Example 3.4, we obtain that (R, m) is a supra compact space, whereas it is not supra semi-Lindelöf. So the converse of Proposition 3.6 fails as well.
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 (X, m) 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.
The proof is similar in case of a supra semi-Lindelöf space.
It can be seen from Example 3.4 that {1, 2} 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 (X, m) 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 semicompactness and the case between parentheses made similarly. Necessity: Let L = {F i : i [ I} be a collection of supra semi-closed subsets of X which has the finite intersection property. Assume that But this contradicts that Λ has the finite intersection property. Thus Λ has, itself, a non-empty intersection. Sufficiency: I} is a collection of supra semi-closed subsets of X which has the finite intersection property.
Proposition 3.12: If A is a supra semi-compact (resp. supra semi-Lindelöf) subset of X and B is a supra semiclosed subset of X, then A > B is supra semi-compact (resp. supra semi-Lindelöf).
Similarly, one can prove the proposition in case of a supra semi-compact space.
Proof: Let g : X Y be a supra semi-irresolute map and let A be a supra semi-compact subset of X.
is supra semi-compact. A similar proof can be given for the case between parentheses.
Theorem 3.14: If g : X Y is a bijective supra semiopen map and Y is a supra semi-compact (resp. supra semi-Lindelöf) space, then X is compact (resp. Lindelöf).
Thus X is compact. A similar proof can be given for the case between parentheses.
Corollary 3.15: If g : X Y is an injective supra semiopen map and g(X) is a supra semi-compact (resp. supra semi-Lindelöf) space, then X is compact (resp. Lindelöf).

Almost supra semi-compact spaces
In this section, the concepts of almost supra semicompact 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 (X, m) 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.   Since Λ has not a finite sub-cover, the supra semi-closure of whose members cover X, then (Q, m) 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.
The converse of Proposition 4.6 is not always true as illustrated in the following example.

Definition 4.9:
A subset E of (X, m) 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 {G i : i [ I} be a supra semi-open cover of a supra semi-clopen subset F of (X, m). Then F c is a supra semi-open set and F # i[I G i . Therefore X = i[I G i < F c . Since (X, m) is almost supra semi-compact, then Hence F is an almost supra semi-compact set.
It can be seen from Example 4.3 that {1, 2} 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 A > B is almost supra semi-compact (resp. almost supra semi-Lindelöf).
is an almost supra semi-compact set. A similar proof can be given for the case between parentheses.
Proof: Let g : X Y be a supra semi-irresolute map and let A be an almost supra semi-compact subset of X. Suppose that {G i : i [ I} be a supra semi-open cover of g(A). This automatically implies that A # i[I g −1 (G i ). Since g is supra semi-irresolute, then g −1 (G i ) is a supra semi-open set, for each i [ I. By hypotheses, A is almost supra semi-compact, then A # i=n i=1 cl m s (g −1 (G i )). It follows, by Theorem 2.6, that cl m 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 g : X Y is a bijective supra semiopen map and Y is almost supra semi-compact (resp. almost supra semi-Lindelöf), then X is almost compact (resp. almost Lindelöf).
Proof: Let {G i : i [ I} be an open cover of X. Then g(X) = g( i[I G i ). Therefore Y = i[I g(G i ). Since g(G i ) is a supra semi-open set, for each i [ I and Y is almost supra semi-compact, then Y = i=n i=1 cl m s (g(G i )). Since g is bijective supra semi-open, then g is supra semi-closed. Therefore by Theorem 2.6, we obtain that cl m s (g(G i )) # g(cl(G i )). Thus X = i=n i=1 cl(G i ). 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 (X, m) satisfies the finite (resp. countable) intersection property, has, itself, a non-empty intersection, then (X, m) 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 (N, m) is the same as in Example 4.2. A collection A n = {n + 1, n + 2, . . .} consists of supra closed subsets of N and has a finite intersection property. Whereas 1 i=1 A n = ∅. So the converse of the above theorem is not always true.

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 (X, m) 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.
The two examples below illustrate that the converse of the above propositions is not always true.
Example 5.4: Let (R, m) be a supra topological space, where μ is the usual topology. From the fact it is connected, we conclude that R and ∅ are the only clopen subsets of (R, m). So it is mildly supra compact. On the other hand, a collection L = {(i, i + 1); i [ Q c } forms a supra semi-open cover of R. Obviously, it is not a countable subcover, hence (R, m) is not mildly supra semi-Lindelöf.
forms a supra semiclopen cover of Q. Since Λ has not a finite subcover, then (Q, m) is not mildly supra semi-compact.
Proof: Let L = {H i : i [ I} be a supra semi-clopen cover of (X, m). Since (X, m) is almost supra semicompact, then A similar proof can be given for the case between parentheses.
The converse of the above corollary need not be true in general as the next example shows. Proposition 5.9: Every supra semi-clopen subset of a mildly supra semi-compact (resp. mildly supra semi-Lindelöf) space (X, m) is mildly supra semi-compact (resp. mildly supra semi-Lindelöf).
Proof: Let {G i : i [ I} be a supra semi-clopen cover of a supra semi-clopen subset F of X. Then F c is a supra semi-clopen set and F # i[I G i . Therefore The proof is similar in case of a mildly supra semi-Lindelöf space.
Proposition 5.10: If A is a mildly supra semicompact (resp. mildly supra semi-Lindelöf) subset of X and B is a supra semi-clopen subset of X, then A > B is mildly supra semi-compact (resp. mildly supra semi-Lindelöf).
Proof: Let L = {G i : i [ I} be a supra semi-clopen cover of A > B. Then 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 (X, m) is mildly supra semi-compact (resp. mildly supra semi-Lindelöf) if and only if every collection of supra semiclopen subsets of X, satisfies the finite (resp. countable) intersection property, has, itself, a non-empty intersection.
Proof: We only prove the theorem when (X, m) is mildly supra semi-compact, the other case can be made similarly. Let L = {F i : i [ I} be a collection of supra semi-clopen subsets of X which has the finite intersection property.
But this contradicts that Λ has the finite intersection property. Thus Λ has, itself, a nonempty intersection.
Conversely, Let {G i : i [ I} be a supra semi-clopen cover of X. Suppose, to the contrary, that{G i : i [ I} has no finite sub-cover. Then X\ i=n i=1 G i = ∅, for any n [ N. Now, i=n i=1 G c i = ∅. This implies that {G c i : i [ I} is a collection of supra semi-clopen subsets of X which has the finite intersection property. Therefore i[I G c i = ∅. Thus X = i[I G i . But this contradicts that {G i : i [ I} is a cover of X. Hence (X, m) is a mildly supra semi-compact space.
Proposition 5.12: If g : X Y is a bijective supra semiopen map and Y is mildly supra semi-compact, then X is mildly compact.
Proof: Let {G i : i [ I} be a clopen cover of X. Then g(X) = g( i[I G i ). Therefore Y = i[I g(G i ). Now, Y is mildly supra semi-compact, then Y = i=n i=1 g(G i ). Since g is bijective supra semi-open, then X = i=n i=1 G i . Hence X is mildly compact.

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.