Local homeomorphism

Remarks

A local homeomorphism is continuous and open. Every homeomorphism is a local homeomorphism, and so is the inclusion of an open subset.

Example: $p \colon \R \to S^1$, $p(t) = (\cos 2\pi t, \sin 2\pi t)$. For $x \in \R$, take $U = (x - 1/2, x + 1/2)$. Then $p(U)$ is the circle minus one point, which is open, and $p|_U$ is a continuous bijection onto it with a continuous inverse given by a branch of the angle.

Non-example: $f \colon \C \to \C$, $f(z) = z^2$. Every neighbourhood $U$ of $0$ contains some $\varepsilon \ne 0$ and $-\varepsilon$, with $f(\varepsilon) = f(-\varepsilon)$, so $f|_U$ is not injective. At every $z \ne 0$, however, $f$ is a local homeomorphism.