Is a monotone union of contractible open sets contractible? (Q340730)

The following very general question is considered.NEWLINENEWLINE{Question.} If a normal space \(X\) is the union of an increasing sequence of open sets \(U_1\subset U_2\subset\dots\) such that each \(U_n\) contracts to a point in \(X\), must \(X\) be contractible?NEWLINENEWLINEThis is motivated by the fact that ``typical constructions of contractible open manifolds produce a space \(X\) that is a union of sets \(U_i\) as indicated.'' Moreover, as is mentioned by the authors, this is usually the first step needed in a construction that yields a contractible open manifold that is not homeomorphic to \(\mathbb{R}^n\).NEWLINENEWLINEThe main results of the paper are:NEWLINENEWLINE{Theorem 1.} If a normal space \(X\) is the union of a sequence of open subsets such that \(\mathrm{cl}(U_n)\subset U_{n+1}\) and \(U_n\) contracts to a point in \(X\) for all \(n\geq1\), then \(X\) is contractible.NEWLINENEWLINE{Corollary 2.} If a locally compact \(\sigma\)-compact normal space \(X\) is the union of an increasing sequence of open sets \(U_1\subset U_2\subset\dots\) such that each \(U_n\) contracts to a point in \(X\), then \(X\) is contractible.NEWLINENEWLINEThe authors list two previously known results, Theorems 3 and 4, which are similar to this one, but which impose stricter conditions on the contractions. Hence the work in this paper provides an improvement upon that of its predecessors. But they are not able to answer the Question.