Give an example of a local homeomorphism $f:X\to Y$ and a subset $A\subset X$ such that $f|_A$ is not a local homeomorphism of $A$ onto $f(A)$.

Question

Give an example of a local homeomorphism $f:X\to Y$ and a subset $A\subset X$ such that $f|_A$ is not a local homeomorphism of $A$ onto $f(A)$.

I am studying algebraic topology and I am faced with this problem that I have no idea how to answer, I would appreciate any help or suggestion, thank you very much.

Answer 1

$f: \mathbf{R} \to S^1$ given by $f(x) = (\cos(x), \sin(x))$ is a covering map hence a local homeomorphism. If $A = [0, 2\pi]$ then at $0$ and $2\pi$ the map $f|_A$ fails to be a local homeomorphism.