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.