Is a monotone union of contractible open sets contractible?

Abstract

This paper presents some partial answers to the following question. QUESTION. If a normal space X is the union of an increasing sequence of open sets U(1), U(2), U(3) ... such that each U(n) contracts to a point in X, must X be contractible? The main results of the paper are: THEOREM 1. If a normal space X is the union of a sequence of open subsets { U(n) } such that the closure of U(n) is contained in U(n+1) and U(n) contracts to a point in U(n+1) for each n > 0, then X is contractible. COROLLARY 2. If a locally compact sigma-compact normal space X is the union of an increasing sequence of open sets U(1), U(2), U(3) ... such that each U(n) contracts to a point in X, then X is contractible.