Comparing spaces by means of 2-homeomorphisms

Topological spaces X and Y are called 2-homeomorphic if there exist homeomorphic closed subspaces of X and Y such that their complements are also homeomorphic. We give some sufficient conditions for spaces to be 2-homeomorphic. In particular, we show that if a space Y is conjugate to a space X, then X and Y are 2-homeomorphic (Theorem 2.1). The complement Rn∖F of an arbitrary compact subset in the Euclidean space Rn is 2-homeomorphic to Rn (Theorem 3.4). Some necessary conditions for two spaces to be 2-homeomorphic are also given. In particular, we capitalize on the next simple fact: if X and Y are nonempty 2-homeomorphic spaces, then some nonempty open subspaces U and V of X and Y, respectively, are homeomorphic (Lemma 2.2; see also Section 4).