Co-Compact Separation Axioms and Slight Co-Continuity

: Via co-compact open sets we introduce co- T 2 as a new topological property. We show that this class of topological spaces strictly contains the class of Hausdorff topological spaces. Using compact sets, we characterize co- T 2 which forms a symmetry. We show that co- T 2 propoerty is preserved by continuous closed injective functions. We show that a closed subspace of a co- T 2 topological space is co- T 2 . We introduce co-regularity as a weaker form of regularity, s-regularity as a stronger form of regularity and co-normality as a weaker form of normality. We obtain several characterizations, implications, and examples regarding co-regularity, s-regularity and co-normality. Moreover, we give several preservation theorems under slightly coc-continuous functions.


Introduction and Preliminaries
Defining a new type of generalized open sets and utilizing it to define new topological concepts is now a very hot research topic [1][2][3][4][5][6][7][8][9][10][11]. As a new type of generalized open sets, Al-Ghour and Samarah in [12] defined coc-open sets as follows: A subset A of a topological space (X, τ) is called coc-open set if A is a union of sets of the form V − C, where V ∈ τ and C is a compact subset of X. Authors in [12] proved that the family of all coc-open sets of a topological space (X, τ) forms a topology on X finer than τ, and via this class of sets they obtained a decomposition theorem of continuity. Research via coc-open sets was and still a hot area of research, indeed authors in [13] introduced and coc-closed, coc-open functions and coccompact spaces, authors in [14] defined new types of connectedness, in [15] authors have studied s-coc-separation axioms, coc-convergence as a new type of convergence for nets and filters was introduced in [16], new types of dimension theory were introduced in [17], in [18] new classes of functions were introduced, and in [19] the suthors generalized co-compact open sets. In this paper, we use coc-open sets to define and investigate new separation axioms and new class of functions.
The material of this paper lies in four chapters. In section two, we define and investigate co-T 2 as a new topological property and as a generalization of T 2 topological property. In section three, we introduce and investigate co-regularity as a weaker form of regularity, s-regularity as a stronger form of regularity and co-normality as a weaker form of normality. In section four, we introduce and investigate slightly coc-continuous functions.
In this paper, for a subset A of a topological space (X, τ), cl τ (A) will denote the closure of A is (X, τ) and τ |A will denote the relative topology of (X, τ) on A.
The following definition and theorem will be used in this sequel: Definition 1. [12] A subset A of a topological space (X, τ) is called co-compact open set (notation: coc-open) if for every x ∈ A, there exists an open set U ⊆ X and a compact subset K of (X, τ) such that x ∈ U − K ⊆ A.

Case 2.
x ∈ U, y ∈ V − K and U ∩ (V − K) = ∅. We have V − K ⊆ X − U and so Since x / ∈ X − U, then x / ∈ cl τ (V − K). ⇐=) Let x, y ∈ X with x = y. By assumption there exist U ∈ τ and a compact set K of (X, τ) such that (x ∈ U − K and y / ∈ cl τ (U − K)) or (x / ∈ cl τ (U − K) and y ∈ U − K).
. Then V ∈ τ and x ∈ V. Therefore, by Theorem 8 we conclude that (X, τ) is co-T 2 .
Since f is injective, then its fibers are compact. Hence, by Lemma This ends the proof according to Theorem 8.
Theorem 11. If (X, τ) is a co-T 2 topological space and A is a closed subset of (X, τ), then (A, τ | A ) is co-T 2 .
Proof. Let x, y ∈ A with x = y. Then x, y ∈ X with x = y. Since (X, τ) is co-T 2 , then there exist Moreover, x ∈ U 1 , y ∈ V 1 and

Co-Regulerity and Co-Normality
We start by the following main definition: Definition 3. A topological space (X, τ) is said to be co-regular if for each closed set F of (X, τ) and each x ∈ X − F there exist U, V ∈ τ k such that x ∈ U, F ⊆ V and U ∩ V = ∅. Theorem 12. If (X, τ) is a topological space such that τ k is the discrete topology on X, then (X, τ) is co-regular.
Proof. Let F be a closed set in (X, τ) and let x ∈ X − F. Let U = {x} and V = F. Then U, V ∈ τ k , x ∈ U, F ⊆ V and U ∩ V = ∅. Therefore, (X, τ) is co-regular.

Corollary 5.
If (X, τ) is a hereditarily compact topological space, then X, τ k is co-regular.

Proof. Theorems 3 and 12
Corollary 6. If X is any non-empty set and τ is the cofinite topology on X, then (X, τ) is co-regular.
Proof. Let (X, τ) be a regular topological space. Let F be a closed set in (X, τ) and let x ∈ X − F. Since (X, τ) is regular, there exist U, V ∈ τ such that x ∈ U, F ⊆ V and U ∩ V = ∅. Since τ ⊆ τ k , it follows that (X, τ) is co-reguler.

Remark 1.
The converse of Theorem 13 is not true in general: By Corollary 6, (R, τ) where τ is the cofinite topology on R is co-regular. On the other hand, it is well known that (R, τ) is not regular.
Proof. (a) Let F be closed in (X, τ) and let x ∈ X − F.
It follows that (X, τ) is co-reguler. (b) Let F be closed in (X, τ k ) and let x ∈ X − F. By (a), we only need to discuss the case when F is not closed in (X, τ). It is not difficult to see that τ k = τ ∪ {U ⊆ X : 1 ∈ U and X − U is finite}. Thus, we must have x = 1 and F is finite.
Suppose to the contrary that (X, τ) is regular. Take x = 3 and F = (0, 2). Then F is closed in (X, τ) with 3 ∈ R − F and by regularity there are U, V ∈ τ such that 3 ∈ U, F ⊆ V and U ∩ V = ∅. Since 1 ∈ F, then 1 ∈ V and so V = R. Thus, 3 ∈ U ∩ V, a contradiction. (d) Let x, y ∈ X with x = y. Without loss of generality we may assume that there are only two cases: It follows that (X, τ) is co-T 2 .

Theorem 15. A topological space (X, τ) is co-regular if and only if for every U
⇐=) Let F be closed in (X, τ) and x ∈ X − F. Then X − F ∈ τ and by assumption there exists Theorem 16. If (X, τ) is a co-regular topological space, then for every closed set F in (X, τ) we have

Theorem 17.
A closed subspace of a co-regular topological space is co-reguler.
Proof. Let (X, τ) be a co-regular topological space and let A be a non-empty closed set of (X, τ). Let x ∈ A and let B be a closed set in (A, It follows that (A, τ| A ) is co-reguler.
Theorem 18. Every s-regular topological space is regular.
The following example shows that the converse of Theorem 18 is not true in general: Example 4. Let X be a set which contains at least three points and let B be a partition on X in which for some B ∈ B, B has at least two points. Let τ be the topology on X having B is a base. Since (X, τ) is zero-dimensional, it is regular. Choose x, y ∈ B such that x = y and let Theorem 19. Let (X, τ) be a T 2 topological space. Then (X, τ) is s-regular if and only if (X, τ) is regular.

Proof. Theorem 18 and Lemma 3
Corollary 7. Every T 3 topological space is s-regular.
Theorem 20. Every normal topological space is co-normal.
Proof. Let (X, τ) be a normal topological space. Let A and B be two disjoint closed sets in (X, τ).
It follows that (X, τ) is co-normal.
is a topological space such that τ k is the discrete topology on X, then (X, τ) is co-normal.
The following example shows that converse of Theorem 20 is not true in general: Example 5. Let X = R and let τ be the cofinite topology. Then Proof. (a) By Corollary 2, τ k is the discrete topology on X. So by Corollary 8, (X, τ) is co-normal.
(b) This is well known in general topology.

Theorem 22. A topological space (X, τ) is co-normal if and only if for every open set U and any closed set
Therefore, (X, τ) is co-normal.
Proof. Let A and B be two disjoint closed sets of (X, τ).

Theorem 23.
A closed subspace of a co-normal topological space is co-normal.
Proof. Let (X, τ) be a co-normal topological space and let A be a non-empty closed set of (X, τ).
Let B, C be two disjoint closed sets in (A, τ| A ). Since A is closed in (X, τ), then B and C are closed in (X, τ). Since (X, τ) is co-normal, then there exist U, This shows that (A, τ| A ) is co-normal. Proof. =⇒) Suppose that (X, τ) is a co-normal topological space. Let U and V be two open sets in (X, τ) with U ∪ V = X. Then X − U and X − V are disjoint closed sets of (X, τ), and since (X, τ) is co-normal, then there exist
Proof. Let (X, τ) be T 1 and co-normal. Let x ∈ X, and F be a closed set in (X,

Slightly Coc-Continuous Functions
Definition 6. [21] A subset A of a topological space (X, τ) is said to be coc-clopen if both A and X − A are coc-open.
It is known that τ k = τ ∪ {U ⊆ X : 1 ∈ U and X − U is finite} (see [12]). Then the set of coc-clopen sets is {U ⊆ X : 1 ∈ U and X − U is finite}. Definition 7. [22] A function f : (X, τ) −→ (Y, σ) is said to be slightly continuous if for each x ∈ X, and for each clopen set V in (Y, σ) containing f (x), there exists an open set U in (X, τ) containing x such that f (U) ⊆ V.

Definition 8.
A function f : (X, τ) −→ (Y, σ) is said to be slightly coc-continuous if for each x ∈ X, and for each clopen set V in (Y, σ) containing f (x), there exists a coc-open set U of (X, τ) containing x such that f (U) ⊆ V.
Theorem 26. Every slightly continuous function is slightly coc-continuous.
Proof. Let f : (X, τ) −→ (Y, σ) be a slightly continuous function. Let V be a clopen set in (Y, σ) such that f (x) ∈ V. Since f is slightly continuous, then there exists U ∈ τ ⊆ τ k containing x such that f (U) ⊆ V. This shows that f is slightly coc-continuous. Theorem 27. [22] Let (X, τ) and (Y, σ) be topological spaces. The following statements are equivalent for a function f : (X, τ) −→ (Y, σ): (a) f is slightly continuous.

Proof.
(a) =⇒ (b): Let V be a clopen set in (Y, σ). Since f is slightly coc-continuous, then for every Moreover, we have It follows that f −1 (V) is coc-closed.
(c) =⇒ (d): Let V be a clopen set in (Y, σ). Then Y − V is also clopen and by (c), f −1 (V) and It follows that f is slightly coc-continuous.  (X, σ). Therefore, f is not slightly continuous.
Example 8. Let R be the real numbers. Take two topologies on R, τ and σ where τ is cofinite topology and σ is discrete topology. Let f : (R, τ) −→ (R, σ) be an identity function. By Corollary 2, τ k is the discrete topology and hence, f is slightly coc-continuous. On the other hand, since {0} is clopen in ( is not open in (R, τ), f is not slightly continuous.
Theorem 29. If f : (X, τ) −→ (Y, σ) is a slightly coc-continuous function and A is a closed set in (X, τ), Thus f |A is slightly coc-continuous.
(b) If f is coc-irresolute and g is slightly continuous, then g • f is slightly coc-continuous.
(c) If f is coc-continuous and g is slightly continuous, then g • f is slightly coc-continuous.
(d) If f is slightly coc-continuous and g is continuous, then g • f is slightly coc-continuous.

Proof. ⇐=)
Let g • f be slightly coc-continuous and V be clopen set in (Z, δ).
. It follows that g is slightly coc-continuous. =⇒) If g is slightly coc-continuous, then by Theorem 30 , g • f is slightly coc-continuous. Definition 10. [21] A topological space (X, τ) is said to be coc-connected if X can not be written as a union of two disjoint non-empty coc-open sets. A topological space (X, τ) is said to be coc-disconnected if it is not coc-connected.
It is clear that a topological space (X, τ) is coc-connected if and only if (X, τ k ) is connected. Therefore, (R, τ) where τ is the co-countable topology on R is an example of a coc-connected topological space.
Also, it is clear that coc-connected topological spaces are connected, however (R, τ) where τ is the cofinite topology on R is an example of a connected topological space that is coc-disconnected.
Proof. Suppose to the contrary that (Y, σ) is disconnected. Then there exist non-empty disjoint open sets U and V such that Y = U ∪ V. It is clear that U and V are clopen sets in (Y, σ). Since f is slightly coc-continuous, then f −1 (U) and f −1 (V) are coc-open in (X, τ). Also, Since f is surjective, f −1 (U) and f −1 (V) are non-empty. Therefore, (X, τ) is coc-disconnected space. This is a contradiction. Corollary 11. The inverse image of a disconnected topological space under a surjective slightly coc-continuous function is coc-disconnected.
Recall that a topological space (X, τ) is called extremally disconnected if the closure of each open set is open.
Proof. Let x ∈ X and let V be an open subset of Y containing f (x). Since (Y, σ) is extremally disconnected, then Cl σ (V) is open and hence clopen. Since f is slightly coc-continuous, then there exists a coc-open set U ⊆ X such that x ∈ U and f (U) ⊆ Cl σ (V). It follows that f is weakly coc-continuous.
Recall that a topological space (X, τ) is called locally indiscrete if every open set is closed.

Proof. Let
V be an open set of Y. Since (Y, σ) is locally indiscrete, then V is clopen. Since f is slightly coc-continuous, then f −1 (V) is coc-clopen set in X. Therefore, f is coc-continuous and contra coc-continuous.
Recall that a topological space (X, τ) is called zero-dimensional if τ has a base consists of clopen sets.
Proof. Let F be a clopen set in Y and let y ∈ Y such that y / ∈ F. Since f is onto then there is x ∈ X such that f (x) = y. Since f is slightly coc-continuous, then by Theorem 28 (c), f −1 (F) is a coc-closed set. Since (X, τ) is s-regular, and x / ∈ f −1 (F), there exist two disjoint open sets U and V such that Therefore, (Y, σ) is clopen regular.
Proof. Let x ∈ X − E. Then f (x) = g(x). Since (Y, σ) is clopen Hausdorff, then there exists two clopen sets V, W ⊆ Y such that f (x) ∈ V, g(x) ∈ W and V ∩ W = ∅. Since f is slightly continuous and g is slightly coc-continuous, then It follows that X − E is coc-open and E is coc-closed.
Proof. Let x, y ∈ X with x = y. Since f is injective, then f (x) = f (y). Since (Y, σ) is T 2 , then there exist two open sets U and V in Y such that f (x) ∈ U, f (y) ∈ V and U ∩ V = ∅. Since (Y, σ) is zero-dimensional, then there exist two clopen sets Since f is slightly coc-continuous, then f −1 (U 1 ) and f −1 (V 1 ) are coc-open sets.
We have This shows that (X, τ k ) is T 2 .
Proof. Let x ∈ X and U be an open set containing x. Since f is an open function, then This implies that (X, τ) is co-regular.
Proof. Let x ∈ X and F be a closed set of X such that x / ∈ F. Since f is a closed function, then f (F) is closed in (Y, σ). Since f is injective, then f (x) / ∈ f (F) and so f (x) ∈ Y − f (F) ∈ σ. Since(Y, σ) is zero-dimensional, then there exists a clopen set V in Y such that f (x) ∈ V ⊆ Y − f (F). Since f is slightly coc-continuous, then f −1 (V) is a coc-clopen set in (X, τ). Moreover, x ∈ f −1 (V) and F ⊆ X − f −1 (V) which is also coc-clopen set of (X, τ). Therefore (X, τ) is co-regular.
Proof. Let A and B be any two non-empty disjoint closed sets in (X, τ). Since f is closed and injective, we have f (A) and f (B) are two disjoint closed sets in Y. Since (Y, σ) is normal, then there exist two open sets U and V in Y such that f (A) ⊆ U, f (B) ⊆ V and U ∩ V = ∅. For each y ∈ f (A), y ∈ U and since (Y, σ) is zero-dimensional there exists a clopen set U y such that y ∈ U y ⊆ U. Thus, f (A) ⊆ ∪{U y : y ∈ f (A)} ⊆ U. Put G = ∪{ f −1 (U y ) : y ∈ f (A)}. Then A ⊆ G ⊆ f −1 (U). Since f is slightly coc-continuous, f −1 (U y ) is coc-open for each y ∈ f (A) and so G = ∪{ f −1 (U y ) : y ∈ f (A)} is coc-open in X. Similarly, there exists a coc-open set H in X such that B ⊆ H ⊆ f −1 (V) and G ∩ H ⊆ f −1 (U ∩ V) = ∅. This shows that (X, τ) is co-normal.

Conclusions
This paper deals with the axioms of separation of points and sets and a type of continuity (Slight coc-continuity). The results are well related to the classical properties. The new concepts are justified by their dependence and independence through examples. In the final part we carry out external characterizations of separation, via slightly coc-continuous functions. The relationship is confronted with zero-dimensional spaces. In future studies, the following topics could be considered: (1) To define other types of sepaeration axioms that are related to co-compact open sets (2) To study the uniform structures that are related to co-compact open sets.
Author Contributions: Formal analysis, investigation, and writing-original draft preparation S.A.G. and E.M. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.