*-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 (Χ,τ) an ideal I on X and A⊆X, ψ(A) is defined as U{Uϵτ:U-AϵI}. A topology, denoted τ*, finer than τ is generated by the basis {U-I:Uϵτ, 1€I), and a topology, denoted, coarser than τ is generated by the basis ψ(τ) = {ψ(U):Uϵτ}. The notation (X,τ,I) denotes a topological space (Χ,τ) with an ideal I on X. A bijection f:(X,τ,I) → (Υ,σ*) is called a *-homeomorphism if f:(X,τ*) → (Υ,σ*) is a homeomorphism, and is called a ψ-homeomorphism if f:(X,) —(Υ,) is a homeomorphism. Properties preserved by *-homeomorphisins arc studied as well as necessary and sufficient conditons for a 0-homeomorphism to be a *-homeomorphism. The semi-homeomorphisms and semi-topological properties of Crossley and Hildebrand [Fund. Math., LXXIV (1972), 233254] are shown to be a special case. © 1990, Hindawi Publishing Corporation. All rights reserved.