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I have found the following problem. Let $A\subseteq X$ be a contractible space. Let $a_0\in A$. Is the embedding $X\setminus A\to X\setminus\{a_0\}$ a homotopy equivalence?

I don't understand the question. I know that two spaces $Y$ and $Z$ are homotopy equivalent when there are such continuous $f:Y\to Z$ and $g:Z\to Y$ that $f\circ g$ is homotopic to $\mathrm{id}_Z$, and $g\circ f$ is homotopic to $\mathrm{id}_Y$. I would think that a homotopy equivalence had to be the pair of functions $f,g$, not just one function. But there is only one function in the problem.

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@ThomasRot OK, thanks. Is there a name for this other function? Like a homotopy inverse or something like this? – Bartek Aug 13 '12 at 15:39
This is precisely the kind of thing that I think the language of category theory helps address quite elegantly. A homotopy equivalence is precisely an isomorphism in a category called the homotopy category ( in the same way that a group isomorphism is an isomorphism in a category called the category of groups and a homeomorphism is an isomorphism in a category called the category of topological spaces... – Qiaochu Yuan Aug 13 '12 at 16:09
up vote 2 down vote accepted

The question asks if there exists another function with the properties you stated. The Nlab calls these maps homotopy inverses of one another.

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