Assume $K\subseteq X$ is a non-empty contractible subspace of a topological space $X$. Let $x\in K$ be any point. I am trying to define an explicit homotopy equivalence between $X \setminus K$ and $X\setminus \{x\}$. How would I go about doing this? Thanks for your help^^
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This is false. Let $X = [0,1], K = [0,1/2]$ and $x = 1/2$.
Paul Frost
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But $0\notin K$? – Margaret Sep 29 '23 at 10:04
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@Margaret My mistake. Example corrected. – Paul Frost Sep 29 '23 at 10:07
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I see, thanks. A generalization of your example (for any $n\geq 1$): $X=[0,1]^n$, $K={1/2}\times [0,1]^{n-1}$ and $x=(1/2,\ldots,1/2)$. – Margaret Sep 29 '23 at 10:15
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This is a follow-up question. Maybe of interest to you. – Margaret Sep 29 '23 at 10:35