Boundary bumping theorem: Let $X$ be a compact connected space, $A$ its closed proper subspace, $C$ a component of $A$, then $C ∩ ∂A ≠ ∅$. ($∂$ means boundary.)
I wonder if the compacness assumption can be relaxed. For example the theorem holds when one assumes that $∂A$ has only finite number of components instead of $X$ being compact. Is there any other variant of the theorem with respect to assumptions? What about counterexamples?
It can't. Consider this connected space $X$ (there's no last vertical line as the limit, so it's not compact):
And take $A$ as the set of all vertical lines along with the point marked by a circle (upper right corner). That's a closed set, the component of the point is just the point itself, but it's not on the boundary of $A$.
The theorem can be generalized in another way. You can leave out that $A$ is closed, then the closure (in $X$) of any component of $A$ intersects the boundary of $A$. (and I believe it can be proven from your version of the theorem)