*-Topological properties. (1990) - Hamlett, T. R.; Rose, David
Abstract : An ideal on a setX is a nonempty collection of subsets ofX closed under the operations of subset (heredity) and finite unions (additivity). Given a topological space( X, τ ) an idealℐ onX andA ⊆ X, ψ ( A ) is defined as⋃ { U ∈ τ : U − A ∈ ℐ } . A topology, denotedτ *, finer thanτ is generated by the basis{ U − I : U ∈ τ, I ∈ ℐ }, and a topology, denoted〈 ψ ( τ ) 〉, coarser thanτ is generated by the basisψ ( τ ) = { ψ ( U ) : U ∈ τ } . The notation( X, τ, ϑ ) denotes a topological space( X, τ ) with an idealℐ onX . A bijectionf : ( X, τ, ℐ ) → ( Y, σ, J ) is called a* -homeomorphism iff : ( X, τ * ) → ( Y, σ * ) is a homeomorphism, and is called aψ -homeomorphism iff : ( X, 〈 ψ ( τ ) 〉 ) → ( Y, 〈 ψ ( σ ) 〉 ) is a homeomorphism. Properties preserved by* -homeomorphisms are studied as well as necessary and sufficient conditions for aψ -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.