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.