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 $U_1\subset U_2\subset U_3\subset \dots$ 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 $cl(U_n)\subset U_{n+1}$ and $U_n$ contracts to a point in $U_{n+1}$ for each $n\ge 1$, then X is contractible.

Corollary 2

If a locally compact σ-compact normal space X is the union of an increasing sequence of open sets $U_1\subset U_2\subset U_3\subset \dots$ such that each $U_n$ contracts to a point in X, then X is contractible.