Let $f: X \rightarrow Y$ be a continuous map between topological spaces.

(a) Assume $X$ is compact and that $Z \subseteq X$ is a closed subset. Prove that $Z$ and $f(Z)$ are both compact.

(b) Suppose that

(i) $f^{-1}(\{y\})$ is compact for each $y \in Y$, and

(ii) if $A$ is any closed subset of $X$, then $f(A)$ is a closed subset of $Y$.

Show that if $K \subseteq Y$ is compact, then $f^{-1}(K)$ is compact.

$\left[\right.$ Hint: Given an open cover $f^{-1}(K) \subseteq \bigcup_{i \in I} U_{i}$, find a finite subcover, say $f^{-1}(\{y\}) \subseteq$ $\bigcup_{i \in I_{y}} U_{i}$, for each $y \in K$; use closedness of $X \backslash \bigcup_{i \in I_{y}} U_{i}$ and property (ii) to produce an open cover of $K$.]