Pseudo-isotopic contractions and compactness (Q1428279)

A topological space \(X\) is called \textit{pseudo-isotopically contractible to} \(x\in X\) \textit{relative to} \(x\) if there is a continuous map \(\Lambda:[0,1]\times X\rightarrow X\) such that \(\Lambda(1,\cdot)=\text{id}_X\), \(\Lambda(0,X)=\{x\}\) and \(\Lambda(t,\cdot)\) is a homeomorphism fixing \(x\) for all \(t\in(0,1)\). The author proves the theorem: Let \(X\) be a locally compact Hausdorff space. Assume that there are elements \(x, y\in X\) such that \(X\setminus\{x,y\}\) is locally connected and \(X\setminus\{y\}\) is pseudo-isotopically contractible to \(x\) relative to \(x\) and \(X\setminus \{x\}\) to \(y\) relative to \(y\). Then \(X\) is compact. Some variations of this theorem are noted, and one corollary obtained is the following: Let \(X\) be a topological \(n\)-manifold, possibly with boundary, and let \(x\) and \(y\) be two points of the interior of \(X\) such that \(X\setminus\{y\}\) is pseudo-isotopically contractible to \(x\) relative to \(x\). Then \(X\approx \mathbb S _n\).