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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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    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 (http://en.wikipedia.org/wiki/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...2012-08-13

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