On Certain Covering Properties and Minimal Sets of Bigeneralized Topological Spaces

We introduce q-Lindelöf, u-Lindelöf, p-Lindelöf, s-Lindelöf, q-countably-compact, u-countably-compact, p-countably-compact, and s-countably-compact as new covering concepts in bigeneralized topological spaces via q-open sets and u-open sets in bigeneralized topological spaces. Relationships between them are studied. As two symmetries relationships, we show that q-Lindelöf and u-Lindelöf are equivalent concepts, and that q-countably-compact and u-countably-compact are equivalent concepts. We focus on continuity images of these covering properties. Finally, we define and investigate minimal q-open set, minimal u-open set, and minimal s-open sets as three new types of minimality in bigeneralized topological spaces.


Introduction
We list the notations used in this paper before the conclusion section. In 1963, Kelly [1] introduced the notion of bitopological spaces as an ordered triple (X, τ, γ) of a set X and two topologies τ, γ, the order is important (i.e., two bitopological spaces (X, τ, γ) and (X, τ , γ ) are identical if, and only if, τ = τ and γ = γ ). After Kelly's initiation of the bitopological notion, many authors generalized many topological concepts to include bitopological spaces. Let X be a nonempty set and σ be a family of subsets of X. Subsequently, σ is called a generalized topology (GT) on X and (X, σ) is a generalized topological space (GTS) [2] if σ contains ∅ and arbitrary union of elements of σ is an element of σ. Let (X, σ) be a GTS and let B ⊆ X. Afterwards, B is called a σ-open set if B ∈ σ. B is called a σ-closed set if X − B is σ-open. (X, µ) is said to be a strong GTS [3] if X ∈ µ. Research in GTS is still a hot area of research in which researchers introduced several types of continuity, compactness, homogeneity, and sets are extended from ordinary topological spaces to include GTSs in [4][5][6][7][8][9][10][11][12][13][14], and others. As a generalization of bitopological spaces, the author in [15] defined bigeneralized topological space, as follows: an ordered triple (X, σ, δ) of a set X and two generalized topologies σ and δ on X is called a bigeneralized topological space (BGTS). A BGTS (X, σ, δ) is said to be strong if (X, σ) and (X, δ) are strong. Throughout this paper, we will assume that all GTSs, as well as BGTSs, are strong.
In [15], Datta defined quasi-open sets in bitopological spaces, as follows: a subset A of a bitopological space (X, τ, γ) is said to be quasi-open if for every x ∈ A there exists U ∈ τ, such that x ∈ U ⊆ A or V ∈ γ, such that x ∈ V ⊆ A. The authors in [16]  In Section 2, we mainly use q-open sets and u-open sets in BGTSs to introduce and study certain covering concepts in BGTSs. Symmetric relationships between them are introduced.
In Section 3, we define two new concepts of continuity between BGTSs. We give several relationships regarding the new continuity concepts and an old continuity concept. We focus on continuity images of covering properties that are defined in Section 2.
In Section 4, we define three types of minimality in BGTSs; we give several relationships regarding them and we focus on continuity images of each of them.

Covering Properties
In this section, we mainly use q-open sets and u-open sets in BGTSs to introduce and study certain covering concepts in BGTSs. Related to the new covering concepts we will give characterizations, implications, and examples.
The family of all q-open sets in (X, σ, δ) will be denoted by q(σ, δ). A subset A of X is said to be quasi-closed (q-closed) if the complement of A is q-open.
(a) A collection F of subsets of X is said to be a σ-cover of X if the union of the elements of F is equal to X. (b) A σ-subcover of a σ-cover F is a subcollection G of F which itself is a σ-cover.  (a) A collection F of subsets of X is called a cover of X if the union of the elements of F is equal to X. (b) A sub-cover of a cover F of X is a subcollection G of F , which itself is a cover of X.
Proof. Suppose (X, σ, δ) is q-Lindelöf and let F be a u-open cover of X. Subsequently, by Theorem 4, F is a q-open cover of X. Since (X, σ, δ) is a q-Lindelöf, F has a countable subcover. Therefore, (X, σ, δ) is u-Lindelöf.
Proof. Let (X, σ, δ) be a u-Lindelöf BGTS and let F be a p-open cover of X. Then F is a u-open cover of X. Since (X, σ, δ) is a u-Lindelöf, F has a countable subcover. Therefore, (X, σ, δ) is p-Lindelöf.
The following two examples will show that the concepts s-Lindelöf and p-Lindelöf are independent, and they will also show that the converse of each of Theorems 7 and 8 is not true in general: (1) (R, µ, σ) is not p-Lindelöf and by Theorem 8, it is not u-Lindelöf: {{x} : x ∈ R} is a p-open cover of R which has no countable sub-cover.
Then (R, µ, σ) is p-Lindelöf but not s-Lindelöf and by Theorem 7, it is not u-Lindelöf.
The converse of Theorem 7 is true for an s-Lindelöf BGTS, as in the following result: Theorem 10. Let (X, σ, δ) be a BGTS. Subsequently, the following are equivalent: Theorem 11. A BGTS (X, σ, δ) is q-countably-compact if and only if it is u-countably-compact.
Proof. Suppose that (X, σ, δ) is q-countably-compact and let F be a countable u-open cover of X. Afterwards, by Theorem 4, F is a countable q-open cover of X. Since (X, σ, δ) be a q-countably-compact, F has a finite subcover. Therefore, (X, σ, δ) is u-countably-compact.
The proof that (X, δ) is δ-countably-compact is similar to that used in the proof of (X, σ) is σ-countably-compact.
Proof. Let (X, σ, δ) be a u-countably-compact BGTS and let F be a countable p-open cover of X. Subsequently, F is a countable u-open cover of X. Because (X, σ, δ) is u-countably-compact, F has a finite sub-cover. Therefore, (X, σ, δ) is p-countably-compact.
The following two examples will show that the concepts s-countably-compact and p-countably-compact are independent, and they will also show that the converse of each of Theorems 12 and 13 is not true in general:  (X, σ, δ) is not p-countably-compact and, by Theorem 13, it is not u-countably compact: {{x} : x ∈ X} is a countable p-open cover of X, which has no finite sub-cover. (2) (X, σ, δ) is s-countably compact: let F be a countable σ-open cover X. Afterwards, X ∈ F and {X} is a finite sub-cover of F . Therefore, (X, σ) is σ-countably-compact. Similarly, we can see that (X, δ) is δ-countably-compact. Proof. (X, σ, δ) is p-countably-compact: let F be a countable p-open cover of X. Since F ∩ (δ − {∅}) = ∅, X ∈ F or O ∈ F . Case 1. X ∈ F . Subsequently, {X} is a finite sub-cover of F and we are done. Case 2. O ∈ F and X / ∈ F . Choose U ∈ F , such that 2 ∈ U. Afterwards, E ⊆ U and so {O, U} is a finite subcover of F . (X, σ, δ) is not s-countably-compact: {E ∪ {x} : x ∈ O} is a countable σ-open cover of X, which has no finite sub-cover. This implies that (X, σ) is not σ-countably-compact and, hence, (X, σ, δ) is not s-countably-compact.
The converse of Theorem 13 is true for s-countably-compact BGTS, as in the following result:  (X, σ) is σ-countably-compact and (X, σ) is σ-countably-compact, in either of the two cases F will have a finite subcover.

Continuity
In this section, we define two new concepts of continuity between BGTSs. We will give several relationships regarding the two new continuity concepts and an old continuity concept. We focus on continuity images of covering properties that are defined in Section 2.

Proof. Follows from Theorems 2 and 16.
The converse of Corollary 1 is not true in general, as can be seen from the following example: (1) g is q-continuous: let H ∈ q(σ 2 , δ 2 ) = δ 2 . Then Proof. Straightforward.
Theorem 18. The q-continuous image of a q-Lindelöf BGTS is q-Lindelöf.
Theorem 19. The s-continuous image of a p-Lindelöf BGTS is p-Lindelöf.
Theorem 20. The q-continuous image of a q-countably-compact BGTS is q-countably-compact.
Theorem 21. The s-continuous image of a p-countably-compact BGTS is p-countably-compact.

Conclusions
In this paper, minimal q-open set, minimal u-open set, and minimal s-open sets, q-Lindelöf, u-Lindelöf, p-Lindelöf, s-Lindelöf, q-countably-compact, u-countably-compact, p-countably-compact, and s-countably-compact as new concepts of BGTSs are introduced and investigated. In future studies, the following topics could be considered: (1) the definition of fuzzy u-open sets and fuzzy q-open sets in fuzzy bigeneralized topological spaces; (2) the extension of the two bitopological concepts quasi N -open sets and Quasi →-open sets, as they appeared in [19,20] to include bigeneralized topological spaces.