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International Journal of Mathematics and Mathematical Sciences
Volume 13, Issue 3, Pages 507-512
http://dx.doi.org/10.1155/S0161171290000734

*-Topological properties

Department of Mathematics, East Central University, Ada 74820, Oklahoma, USA

Received 12 June 1989; Revised 4 December 1989

Copyright © 1990 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 on X and AX, ψ(A) is defined as {Uτ:UA}. A topology, denoted τ*, finer than τ is generated by the basis {UI:Uτ,I}, and a topology, denoted ψ(τ), coarser than τ is generated by the basis ψ(τ)={ψ(U):Uτ}. The notation (X,τ,ϑ) denotes a topological space (X,τ) with an ideal on X. A bijection f:(X,τ,)(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.