We informally say that a space has a hole if it’s identity map is not nullhomotopic, and classify different kinds of holes in a space with homotopy groups. However, I was wondering if there is some to say that a hole is created by removing some subset from a space (taking the subspace topology on its relative complement). I think it can’t be reduced to the properties of the original space and the space with the set removed in isolation (I’m picturing puncturing a hole in an “infinite-holed solid torus” to obtain a set homeomorphic to the original), so a definition must directly or indirectly refer to the inclusion mapping between them. My intuition is that a hole is an obstruction that prevents some topological structure in a space from being moved somewhere else, so my current idea for a definition is this:
In a topological space $X$, the removal of $Y\subseteq X$ creates a hole if and only if, there exist $f,g$ with codomain $X$ and ranges that are subsets of $X\setminus Y$, such that $f$ and $g$ are homotopic in $X$ but are not homotopic in $X\setminus Y$ equipped with the subspace topology from $X$.