On topological groups via a-local functions

Abstract

An ideal on a set X is a nonempty collection of subsets of X which satisfies the following conditions (1)A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. Given a topological space (X; ) an ideal I on X and A ⊂ X, ℜa(A) is defined as ∪{U ∈ a : U − A ∈ I}, where the family of all a-open sets of X forms a topology [5, 6], denoted by a. A topology, denoted a , finer than a is generated by the basis (I; ) = {V − I : V ∈ a(x); I ∈ I}, and a topology, denoted ⟨ℜa( )⟩ coarser than a is generated by the basis ℜa( ) = {ℜa(U) : U ∈ a}. In this paper A bijection f : (X; ; I) → (X; ;J ) is called a A∗-homeomorphism if f : (X; a ) → (Y; a ) is a homeomorphism, ℜa-homeomorphism if f : (X;ℜa( )) → (Y;ℜa()) is a homeomorphism. Properties preserved by A∗-homeomorphism are studied as well as necessary and sufficient conditions for a ℜa-homeomorphism to be a A∗-homeomorphism.

Keywords

• ℜa-homeomorphism, topological groups, a-local function, ideal spaces, ℜa-operator, A∗-homeomorphism