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