Is a monotone union of contractible open sets contractible?
  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. 
Fredric D. Ancel and Robert D. Edwards
math.GN math.AT math.GT