A Schröder-Bernstein construction of homeomorphisms of P-spaces

Abstract

Unlike constructing bijections or measurable isomorphisms, there seems no general "simple" analogous  method to construct  a homeomorphism with given topologies. In particular, we show that under some formally familiar  conditions, the classical  Schröder-Bernstein construction idea also gives homeomorphisms of spaces where every $G_\delta$ set is open.