Quasi-homeomorphisms and lattice-equivalences of topological spaces

Extract

In his paper [1], Thron introduced a concept of lattice-equivalence of topological spaces. Let C(X) denote the lattice of all closed sets of a topological space X. Two topological spaces X and Y are said to be lattice-equivalent if there exists a lattice-isomorphism between C(X) and C(Y). It is clear that for any continuous function f: X → Y, the induced map ψ_f: C(Y) → C(X), defined by ψ(F)=f⁻¹(F), is a lattice-homomorphism. Furthermore, if h: X→ Y is a homeomorphism then ψ_h: C(Y) → C(X) is a lattice-isomorphism. Thron proved among others that for T_D-spaces X and Y, any lattice-isomorphism: C(Y) → C(X) can be induced by a homeomorphism f: X → Y in the above way.