Almost Homeomorphisms on Bigeneralized Topological Spaces

The aim of this paper to introduce the concept of almost (μ, μ ′ )(m,n)homeomorphism on bigeneralized topological space. Also, we introduce the concept of (μ, μ ′ )(m,n)-homeomorphism on bigeneralized topological space. Basic properties, characterizations and relationships between (μ, μ ′ )(m,n)-homeomorphism and almost (μ, μ ′ )(m,n)-homeomorphism are obtained.


introduction
Á. Csaŕszaŕ [3] introduced the concepts of generalized neighborhood systems and generalized topological spaces.He also introduced the concepts of continuous functions and associated interior and closure operators on generalized neighborhood systems and generalized topological spaces.In particular, he investigated characterizations for the generalized continuous function by using a closure operator defined on generalized neighborhood systems.In [4] , he introduced and studied the notions of g-α-open sets, g-semi-open sets, gpreopen sets and g-β-open sets in generalized topological spaces.W. K. Min [6] introduced the notion of almost (g, g )-continuity and investigated properties of such functions and relationships among (g, g )-continuity, almost (g, g )continuity and weak (g, g )-continuity.C. Boonpok [1] introduced the concept of bigeneralized topological spaces and studied (m.n)-closed sets and (m.n)open sets in bigeneralized topological spaces.He also introduced the notion of weakly open functions on bigeneralized topological spaces and investigated properties of such functions.Recently, in 2011, T. Duangphui and at al [5] introduced the notions of (μ, μ ) (m,n) -continuous, almost (μ, μ ) (m,n) -continuous and weakly (μ, μ ) (m,n) -continuous functions.They obtained many characterizations and properties of such functions.In this paper, we introduce the concepts of (μ, μ ) (m,n) -homeomorphism and almost (μ, μ ) (m,n) -homeomorphism on bigeneralized topological spaces.We obtain several characterizations and properties of almost (μ, μ ) (m,n) -homeomorphism.Moreover, many terms are reduced when we use the term of pairwise almost (μ, μ ) (m,n) -homeomorphism instead of the term of almost (μ, μ ) (m,n) -homeomorphism as we see through this paper.

Preliminaries
Definition 2.1.[3] Let X be a nonempty set and μ a collection of subsets of X.Then μ is called a generalized topology (briefly GT ) on X if and only if ∅ ∈ μ and V i ∈ μ for i ∈ I = ∅ implies i∈I V i ∈ μ.We call the pair (X, μ) a generalized topological space (briefly GT S) on X.The elements of μ are called μ-open sets and the complements are called μ-closed sets.
The closure of a subset A in a generalized topological space (X, μ), denoted by c µ (A), is the intersection of generalized closed sets including A, i.e., the smallest μ-closed set containing A. The interior of A, denoted by i µ (A), is the union of generalized open sets contained in A, i.e., the largest μ-open set contained in A.
Let (X, μ 1 , μ 2 ) be a bigeneralized topological space and A a subset of X.The closure of A and the interior of A with respect to μ m are denoted by c µm (A) and i µm (A), respectively, for m = 1, 2.
From the above definitions of (μ, μ ) (m,n) -homeomorphism and almost (μ, μ ) (m,n)homeomorphism, we have the following implication but the reverse relation may not be true in general: Proof.Obvious from Definition 2.6 and Definition 3.2.

Definition
subset H of X and for every μ (m,n) -regular closed subset F of Y , where m, n = 1, 2 and m = n.Proof.(1) =⇒ (2) Let U be any μ (m,n) -regular open subset of X, where m, n = 1, 2 and m = n.By Lemma 3.2(4), we have c µ open subset of X for every μ (n,m) -regular open subset B of Y , where m, n = 1, 2 and m = n ; (4) For every subset A of X, where m, n = 1, 2 and m = n, we have regular open subset of Y , where m, n = 1, 2 and m = n.By the proof above it follows A is μ (m,n) -regular open subset of X.(3) =⇒ (1) Let A = f −1 (B) be any μ (m,n) -regular open subset of X, where m, n = 1, 2 and m = n.By (3), f (A) = B is μ (n,m) -regular open subset of Y and hence f (A) is μ n Y -open subset of Y .By Lemma 3.2(6), f −1 is pairwise almost (μ, μ ) (m,n) -continuous.Let B be μ (m,n) -regular open subset of Y , where m, n = 1, 2 and m = n.By (3),