On h-open sets and h-continuous functions

In this paper, we introduce a new class of open sets in a topological space (X, τ) called h-open sets. Also, introduce and study topological properties of h-interior, h-closure, h-limit points, h-derived, h-interior points, h-border, h-frontier and h-exterior by using the concept of h-open sets. Moreover introduce the notion of h-continuous functions, h-open functions, h-irresolute functions, h-totally continuous functions, h-contra-continuous functions, h-homeomorphism and investigate some properties of these functions and study some properties, remarks related to them.


Introduction and Preliminaries
The concept of open sets is an important concepts in topology and its applications.Levine [7] introduced semi-open set and semicontinuous function, Njastad [8] introduced α-open set, Askander [15] introduced iopen set, iirresolute mapping and i-homeomorphism, Biswas [6] introduced semi-open functions, Mashhour, Hasanein, and El-Deeb [1] introduced α-continuous and α-open mappings, Noiri [16] introduced totally (perfectly) continuous function, Crossley [11] introduced irresolute function, Maheshwari [14] introduced α-irresolute mapping, Beceren [13] introduced semi α-irresolute functions, Donchev [4] introduced contra continuous functions, Donchev and Noiri [5] introduced contra semi continuous functions, Jafari and Noiri [12] introduced Contra-α-continuous functions, Ekici and Caldas [3] introduced clopen-T 1 , Staum [10] introduced, ultra hausdorff, ultra normal, clopen regular and clopen normal, Ellis [9] introduced ultra regular, Maheshwari [13] introduced s-normal space, Arhangel [2] introduced α-normal space.For a subset A of a topological space (X, τ ), the closure of A, the interior of A with respect to τ are denoted by Cl(A) and Int(A) respectively.The complement of A is denoted by A c .A subset A of a topological space (X, τ ) is said to be clopen set, if A is open and closed.This work consists of two sections.In section one we will introduce and study a new class of open sets which is called h-open set and introduce the notions of h-interior, h-closure, h-limit points, h-derived, h-interior points, h-border, h-frontier and h-exterior by using the concept of h-open sets, and study their topological properties.In section two we will present the notion of h-continuous functions, h-open functions, h-irresolute functions, h-totally continuous functions, h-contracontinuous functions, h-homeomorphism and investigate some properties of these functions and study some properties, remarks related to them.

topological space and let A ⊆ X. The hinterior of A is defined as the union of all h-open sets in X content in A, and is denoted by Int
2. Int h (A) ⊆ A.

A is h-closed if and only if
Definition 2.4.Let (X, τ ) be a topological space and let A ⊆ X.A point x ∈ X is said to be h-limit point of A if it satisfies the following assertion:

The set of all h-limit points of A is called the h-derived set of A and is denoted by D h (A).
Note that for a subset A of X, a point x ∈ X is not a h-limit point of A if and Theorem 2.3.Let (X, τ ) be a topological space and let A be a subset of X.Then the following are equivalent This is a contradiction, and hence (2) is valid.
Theorem 2.4.Let (X, τ ) be a topological space and let A ⊆ B ⊆ X.Then 3), ( 4) and ( 5) the proof is easy.Theorem 2.5.Let τ 1 and τ 2 be topologies on X such that τ h 1 ⊆ τ h 2 .For any subset A of X, every h-limit point of A with respect to τ 2 is a h-limit point of A with respect to τ 1 .

A is h-closed if and only if D
Proof.Let x be a h-limit point of A with respect to Hence x is a h-limit point of A with respect to τ 1 .
Remark 2.3.The converse of the Theorem.2.5, need not be true as shown in the following example.
Proof.Straightforward.Theorem 2.7.If A is a subset of a discrete topological space (X, τ ), then D h (A) = Ø.
Proof.Let x ∈ X. Recall that every subset of X is open, and so h-open.In particular, the singleton set G = {x} is h-open.But x ∈ G and G ∩ A = {x} ∩ A ⊆ {x}.Hence x is not a h-limit point of A, and so D h (A) = Ø.Theorem 2.8.Let (X, τ ) be a topological space and let A, B subsets of X.

topological space and let A subset of X. Then A is h-open if and only if there exists an open set U in X such that A ⊆ U ⊆ Cl(A).
Proof.Straightforward.

Lemma 2.2. The intersection of an open set and a h-open set is a h-open set.
Proof.Let A be an open set in X and B a h-open set in X.Then there exists an open set U in X such that B ⊆ U ⊆ Cl(B).It follows that

is called the h-border of A, and the set F r h (A) = Cl h (A)\Int h (A) is called the h-frontier of A. Note that if A is a h-closed subset of X, then b
Theorem 2.9.Let (X, τ ) be a topological space and let A ⊆ X.Then 8) Applying (7) and Theorem.2.4 (1), we have b h

Proof. (1) and (2). Straightforward. (3) Since Int
Lemma 2.3.Let (X, τ ) be a topological space and let A ⊆ X.Then A a h-closed if and only if F r h (A) ⊆ A.

Proof. Assume that
Theorem 2.10.Let (X, τ ) be a topological space and let A ⊆ B ⊆ X.Then Proof.Let f : (X, τ ) → (Y, σ) be h-continuous and g : (Y, σ) → (Z, η) be continuous .Let U be an open set in Z. Since, g is continuous, then

Remark 3 . 1 .
The converse of the Theorem.3.1, need not be true as shown in the following example.