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Title:*-Topological properties
Author:T. R. Hamlett ; David Rose
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.
Journal:International Journal of Mathematics and Mathematical Sciences
Issn:01611712
EIssn:16870425
Year:1990
Volume:13
Issue:3
pages/rec.No:507-512
Key wordsideal ; regular open ; semi-open ; semi-homeomorphism ; semi-topological property ; semiregular ; compatible ideal ; topological property ; *-topological property ; τ-boundary ideal ; nowhere dense sets ; meager sets.
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