An ideal on a set is a nonempty collection of subsets of closed under the operations of subset (heredity) and finite unions (additivity). Given a topological space an ideal on and , is defined as . A topology, denoted , finer than is generated by the basis , and a topology, denoted , coarser than is generated by the basis . The notation denotes a topological space with an ideal on . A bijection is called a -homeomorphism if is a homeomorphism, and is called a -homeomorphism if is a homeomorphism. Properties preserved by -homeomorphisms are studied as well as necessary and sufficient conditions for a
-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.