2016/06/16 by Ancel, Fredric D., Edwards, Robert D.
#54D99 #55M99 #55P99 #57N99 #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1606.05379
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.