* -Topological properties

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.