*-Topological properties

Abstract

An ideal on a set X is a nonempty collection of subsets of X closed under the operations of subset (heredity) and finite unions (additivity). Given a topological space (X,τ) an ideal ℐ on X and A⊆X, ψ(A) is defined as ⋃{U∈τ:U−A∈ℐ}. A topology, denoted τ*, finer than τ is generated by the basis {U−I:U∈τ,I∈ℐ}, and a topology, denoted 〈ψ(τ)〉, coarser than τ is generated by the basis ψ(τ)={ψ(U):U∈τ}. The notation (X,τ,ϑ) denotes a topological space (X,τ) with an ideal ℐ on X. A bijection f:(X,τ,ℐ)→(Y,σ,J) is called a *-homeomorphism if f:(X,τ*)→(Y,σ*) is a homeomorphism, and is called a ψ-homeomorphism if f:(X,〈ψ(τ)〉)→(Y,〈ψ(σ)〉) is a homeomorphism. Properties preserved by *-homeomorphisms are studied as well as necessary and sufficient conditions for a ψ -homeomorphism to be a *-homeomorphism. The semi-homeomorphisms and semi-topological properties of Crossley and Hildebrand [Fund. Math., LXXIV (1972), 233-254] are shown to be special case.

Keywords

• ideal
• regular open
• semi-open
• semi-homeomorphism
• semi-topological property
• semiregular
• compatible ideal
• topological property
• *-topological property
• τ-boundary ideal
• nowhere dense sets
• meager sets.