Sequences of homeomorphisms which converge to homeomorphisms

Abstract

A technique often used in topology involves the inductive modification of a given mapping in order to achieve a limit mapping having certain prescribed properties. The following definition will facilitate the discussion. Suppose X and Y are topological spaces, and {Wi}, = 1, 2, , is a countable collection of subsets of X. Then a sequence {/*}, i ^ 0, of mappings from X into Y is called stable relative to {Wi} if fi\(X-Wi) = /*-! I(X -Wi),i, = 1, 2, . Note, in the above definition, that if {Wi} is a locally finite collection, then liii-oo/ is necessarily a well defined mapping from X into Y, and is continuous if each /; is continuous. In a typical smoothing theorem, a C