Let $X$ be a topological space and $N$ a subset of $X$. Is it true that the closure of
$N$ in $X$ is homotopy equivalent to $N$. I think it is not. take for example $N=\mathbb Q\subset \mathbb R=X$. Then $\bar Q=\mathbb R$ is contractible while $\mathbb Q$ is not even connected. This question came to my mind when i read that a submanifold with boundary keeps its homotopy type after removing its boundary.
Are there conditions where the closure of a subset has the homotopy type of the subset?
Thank you in advance.