Is a monotone union of contractible open sets contractible?

TL;DR

This paper investigates whether a normal space formed by an increasing union of contractible open sets must itself be contractible, providing partial affirmative answers under specific conditions.

Contribution

It establishes that if each open set contracts to a point within the next, then the union space is contractible, extending known results to broader classes of spaces.

Findings

If each open set contracts within the next, the union is contractible.

The result applies to locally compact sigma-compact normal spaces.

Under certain closure conditions, the union of contractible open sets is 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.