Example 4.1:

(i) Suppose $X$ is a compact space and $Y$ is a Hausdorff space then any surjective continuous map $f : X \longrightarrow Y$ is a closed map.

(ii) The reader may check that $\phi : \mathbb{R} \longrightarrow S^1$ given by $\phi(t) = \exp(2\pi it)$ is an open mapping.

(iii) The map $\phi:[0, 1]\longrightarrow S^1$ given by $\phi(t) = \exp(2\pi it)$ is closed but not open.