Let $X$ be a topological $n$-manifold and $N$ a $d$-submanifold of $X$, ($d\leq n$), then under what conditions on $X$ and $N$ do we have that the complement $X-N$ is again a manifold and what is the dimension of the submanifold $X-N$?
For example if $N$ is a submanifold that is closed as a subset of $X$ then $X-N$ is an open subset of $X$ hence it is a submanifold of $X$ with the same dimension $n$. but what about the case of $N$ not being a closed subset of $X$?