Some new higher separation axioms via sets having non-empty interior

Some new higher separation axioms called normal, -normal, and -normal are introduced by using a newly defined closure operator called -closure operator and interrelation among these variants of normality are established. It is also observed that the class of -normal spaces contains all these variants of normality.


Introduction
have been introduced and utilized to study general topology in the past. Interior of these closed sets (or generalized closed sets) need not be non-empty, whereas interior of a non-empty regularly closed (or canonically closed) set is always non-empty. In this paper, some higher separation axioms are introduced and studied by separating sets with non-empty interior which are obtained through a generalized closure operator called ⋆ -closure operator. It is evident from Smyth (1995) that topological structures that are more general than the usual topology are worthy of study because they can provide a suitable framework for various approaches to digital topology. Cech (1966) introduced Cech closure operator which are obtained from the Kuratowski ones by omitting the requirement of idempotency. Liu (2010), studied

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In the topological literature several forms of closed sets exist which has been utilized by researchers to study various topological properties. Interior of these closed sets need not be non-empty, whereas interior of a non-empty regularly closed (or canonically closed) set is always non-empty. In this paper, some higher separation axioms are introduced and studied by separating sets with non-empty interior which are obtained through a generalized closure operator. It is observed that this class of space is independent of the class of normal topological spaces. In contrary to normality, it is shown that this class of space neither contains the class of compact Hausdorff spaces nor contains the class of metrizable spaces.
representation of closure concepts via relations and established a one to one correspondence between quasi-discrete closures (Galton, 2003;Slapal, 2003) and reflexive relations. In this paper, the authors introduced some new higher separation axioms including ⋆-normal spaces and it is observed that this class of space is independent of the class of normal topological spaces. In contrary to normality, it is shown that the class of ⋆-normal spaces neither contains the class of compact Hausdorff spaces nor contains the class of metrizable spaces. Further, it is obsereved in Das (2013) that the class of -normal spaces contains several generalizations of normality such as densely normal spaces (Arhangel'skii, 1996), Δ-normal spaces (Das, 2009), quasi normal spaces (Kalantan, 2008), -normal spaces (Kalantan, 2008), and almost normal spaces (Singal & Arya, 1970). In this paper, we have shown that the important class of -normal spaces also contains the class of ⋆-normal spaces which is independent of normality. Throughout the present paper no separation axiom is assumed unless stated otherwise. Interior of a set A in X is either denoted by A o or int X (A). Similarly, closure of a set A in X is denoted by cl X (A) or Ā .

Preliminaries
Definition 2.1 Let X be any topological space, then an operator ⋆ : It can be easily verified that ⋆-closure operator satisfies the following axioms: For any topological space if a set satisfies Case I, then the set is regularly closed (Kuratowski, 1958), if it satisfies Case II, then the set is semi-open (Levine, 1963) (semi-open sets are referred as -sets in Njastad 1965) and sets satisfying Case III are known as pre-closed sets (Mashhour, Abd El-Monsef, & El-Deeb, 1982).
Definition 2.4 (Mashhour et al., 1982) A subset A of a topological space X is said to be pre-closed if Complement of a ⋆-closed set is said to be ⋆-open. Definition 2.6 Let X be a topological space and A be a subset of X. Then a point x in X is said to be ⋆ -limit point of A if for every open neighborhood U x around x there exists an non-empty open set V in X such that V ⊆ A and V ∩ U x is not empty.
Remark 2.7 Let X be a topological space and A ⊆ X then A is ⋆-closed iff A contains all its ⋆-limit points. Every ⋆-limit point of a set A is a limit point of the set but the converse need not to be true. From the above definition it is obvious that every ⋆-closed set is pre-closed but the converse need not to be true. Further every closed set is pre-closed but ⋆-closed sets are independent of closed sets.
The following implications are obvious from the definitions.
Example 2.9 A set which is pre-closed but not ⋆-closed. Let X = {a, b, c} and X = {X, , {b, c}} be the topology on X, then A = {a, b} is a pre-closed set which is not ⋆-closed in X as int(A) = .
Example 2.10 A set which is closed but not ⋆-closed.
Example 2.11 A set which is ⋆-closed but not closed.
Proposition 2.13 Let X be a topological space and A and B be two ⋆-closed sets in X then A ∪ B is ⋆ -closed.
Remark 2.14 Intersection of two ⋆-closed sets need not be ⋆-closed in X. Let X be the real line with usual topology and let A = [0, 1] and B = [1, 2]. Then A and B are regularly closed, thus ⋆-closed but

Some higher separation axioms
Definition 3.1 A topological space X is said to be (1) ⋆-normal if every pair of disjoint ⋆-closed sets can be separated by disjoint open sets.
(2) strongly ⋆-normal (s ⋆ -normal) if every pair of disjoint pre-closed sets can be separated by disjoint open sets.
(3) weakly ⋆-normal (w ⋆ -normal) if every pair of disjoint closed sets whose interior is non-empty can be separated by disjoint open sets.
Recall that a space X is completely normal if every pair of separated sets can be separated by disjoint open sets and a space X is -normal (Stchepin, 1972) (mildly normal (Singal & Singal, 1973)) if every pair of disjoint regularly closed sets can be separated by disjoint open sets.
Since every regularly closed set is ⋆-closed and closure of every open set is regularly closed, the proof of the following theorem is analogous to the proof of Urysohn's lemma. 2 ) and f −1 ( 1 2 , 1] are two disjoint open sets separating A and B, respectively. Hence X is ⋆-normal. □ It is well known that every metrizable space is normal and every compact Hausdorff space is normal, but the following example establishes that these facts need not hold for ⋆-normal spaces.
Proof Let A be a pre-closed set in X and U be a pre-open set containing A. Then A and X − U are disjoint pre-closed sets in X and so by s ⋆ -normality of X, there exist two disjoint open sets V and W such that Proof Let X be a w ⋆ -normal space. Let A be a closed set with intA ≠ and U be an open set containing A satisfying U ≠ X. Here A and B = X − U are two disjoint closed sets whose interiors are non-emp- The converse part is obvious. □ The proof of the following theorem is analogous to the proof of Urysohn's type lemma. The following implications are obvious from the definitions defined in the present paper and notions defined in Arhangel'skii (1996), Das (2009Das ( , 2013, Kalantan (2008), Singal and Arya (1970) but none of these is reversible.  b, c}, {d, e}, {c, d, e}, {a, d, e}, {a, b, c, d}, {b, d, e}, {a, c, d, e}} and so every pair of disjoint ⋆-closed sets can be separated by disjoint open sets. Hence X is ⋆-normal.
Example 3.10 A space which is ⋆-normal but not s ⋆ -normal.
Let X be the space as defined in Example 3.9. This space is not s ⋆ -normal because for disjoint pre- Example 3.12 A space which is w ⋆ -normal but not normal. Example 3.9 is w ⋆ -normal but not normal because for disjoint closed sets {a} and {c} there does not exist disjoint open sets separating them.
Example 3.13 A space which is normal but not s ⋆ -normal.
Example 3.11 is normal but not s ⋆ -normal because for disjoint pre-closed sets {a, b} and {c, d} there does not exist disjoint open sets separating them.
Example 3.14 A space which is -normal but not w ⋆ -normal.
Let X = {a, b, c, d, e, f } and be the topology generated by the subbasic open sets {{a}, {a, b, c}, {a, b, c, d}, {e}, {c, d, e}, {b, c, d, e}, {f }}. Here regularly closed sets are {{a}, {b, c, d, e, f }, {e, f }, {a, b, c, d}, {a, b, c, d, f }, {e}, {a, f }, {b, c, d, e}, {a, b, c, d, e}, {f  Recall that a subset A of a space X is locally closed if A is open in its closure in X. A space X is a submaximal space (Bourbaki, 1961) if every subset of X is locally closed. A space X is a door space (Arhangel'skii & Collins, 1995)

⋆-Closure operator in lower separation axioms
It is well known that in a T 1 -space every singleton set is closed. Also it is observed in Kohli and Das (2002) that a space is Hausdorff if and only if every singleton set is -closed. But in contrary to this the following result establishes that if in a space every singleton is ⋆-closed then the space is discrete, thus satisfies all separation axioms.

Theorem 4.1 A topological space X is discrete if and only if every singleton in X is ⋆-closed.
Proof Let X be a discrete space and let x ∈ X. Since X is discrete, every singleton is open. Thus int{x} ≠ and ⋆{x} ⊆ {x}. Hence {x} is not only ⋆-closed but also regularly closed. Conversely, let every singleton in X is ⋆-closed. Then interior of every of singleton set {x} in X is non-empty and so int{x} = {x}. Hence X is discrete. □ Definition 4.2 A topological space X is said to be a T ⋆ 1 space if for every distinct points x and y there exist ⋆-open sets U and V such that x ∈ U, y ∉ U and y ∈ V, x ∉ V.
Definition 4.3 A topological space X is said to be a T ⋆ 2 space if for every distinct points x and y there exist disjoint ⋆-open sets U and V such that x ∈ U and y ∈ V.

From the definitions it is obvious that every
Proof Let X be a T 2 space and let x, y be two distinct points in X. Then there exist two disjoint open sets U and V such that x ∈ U and y ∈ V.
Thus, by Proposition 2.12, U and V are two disjoint ⋆-open sets in X separating x and y, respectively. Hence X is T ⋆ 2 . □ From definitions and above observation the following implications are obvious. Remark 4.6 From the above example it is clear that T ⋆ 1 spaces need not imply T 1 -ness. Thus it is natural to ask : Does T 1 -ness imply T 1 ⋆-ness? The following example provide answer to this question in negative. Thus the notions T 1 and T 1 ⋆ are independent of each other.
Let X be an infinite set with co-finite topology.
Clearly the space is T 1 . But the space is not T ⋆ 1 as there does not exists any ⋆-open set in X. Suppose there exists a ⋆-open set U ≠ X in X containing x then there is a point y ∉ U. Thus X − U is a ⋆-closed set containing y. Since X − U is ⋆-closed, int(X − U) ≠ . Thus X − U contains infinitely many points and so x ∉ (X − U) will become a limit point of (X − U). Hence (X − U) cannot be ⋆-closed and so U cannot be ⋆-open.

Subspaces and preservation under mapping
Let X be a topological space and Y be its subspace then the following example shows that every ⋆-closed subset A of Y need not be ⋆-closed in X.
Example 5.1 Let X = {a, b, c, d} and X = { , X, {a}, {c}, {a, c}, {a, c, d}} Proposition 5.2 Let X be a topological space and Y be its ⋆-closed subspace, then every ⋆-closed subset of Y is ⋆-closed in X. Proof Proof Let X be a ⋆-normal space and Y be its ⋆-closed subspace. To show that Y is ⋆-normal, let A and B be two disjoint ⋆-closed sets in Y. Since, Y is ⋆-closed subspace of X then A and B are also two disjoint ⋆-closed sets in X. So by ⋆-normality of X there exist two disjoint open sets say U and V in X Let X be a topological space and Y be its subspace then every closed subset A of Y with int Y (A) ≠ � need not be closed in X is shown in the following example. Theorem 5.5 Let X be a topological space and Y be its closed subspace with int X (Y) ≠ �, then every Proof Let X be a w ⋆ -normal space and Y be a closed subspace of X with int X (Y) ≠ �. To show that Y is w ⋆ -normal, let A and B be two disjoint closed sets in Y with int Y (A) ≠ � and int Y (B) ≠ �. Since Y is closed subspace of X and int X (Y) ≠ �, A and B are also two disjoint closed sets in X and int X (A) ≠ �, int X (B) ≠ �. So by w ⋆ -normality of X there exist two disjoint open sets say U and V in X such that A ⊆ U and B ⊆ V. Clearly, U ∩ Y and V ∩ Y are two disjoint open sets in Y such that A ⊆ U ∩ Y and B ⊆ V ∩ Y. Thus, Y is a w ⋆ -normal space. □ Let X be a topological space and Y be its subspace then every pre-closed subset A of Y need not be pre-closed in X is evident from Example 5.4, in which A is preclosed in Y but not pre-closed in X.
Theorem 5.7 Let X be a topological space and Y be its pre-closed subspace, then every pre-closed subset of Y is pre-closed in X.
Theorem 5.8 A pre-closed subspace of a s ⋆ -normal space is s ⋆ -normal.
Proof Let X be a s ⋆ -normal space and Y be its pre-closed subspace. To show that Y is s ⋆ -normal, let A and B be two disjoint pre-closed sets in Y. Since Y is pre-closed subspace of X, A and B are also two disjoint pre-closed sets in X. So by s ⋆ -normality of X, there exist two disjoint open sets say U and V in X such that A ⊆ U and B ⊆ V. Clearly, U ∩ Y and V ∩ Y are two disjoint open sets in Y such that A ⊆ U ∩ Y and B ⊆ V ∩ Y. Thus Y is a s ⋆ -normal space. □ Theorem 5.9 A continuous clopen image of a ⋆-normal space is ⋆-normal.
Proof Let X be a ⋆-normal space and f :X → Y be a continuous, closed and onto map. To show that Y is ⋆-normal, let A and B be two disjoint ⋆-closed sets in Y. Then f −1 (A) and f −1 (B) are disjoint ⋆-closed sets in X. Since X is ⋆-normal, there are two disjoint open sets U and V in X such that f −1 (A) ⊆ U and f −1 (B) ⊆ V. Also, since f is closed map, f (X ⧵ U)and f (X ⧵ V) are closed in Y. □