Let $f: X \rightarrow Y$ be a map between metric spaces. Prove that the following two statements are equivalent:

(i) $f^{-1}(A) \subset X$ is open whenever $A \subset Y$ is open.

(ii) $f\left(x_{n}\right) \rightarrow f(a)$ for any sequence $x_{n} \rightarrow a$.

For $f: X \rightarrow Y$ as above, determine which of the following statements are always true and which may be false, giving a proof or a counterexample as appropriate.

(a) If $X$ is compact and $f$ is continuous, then $f$ is uniformly continuous.

(b) If $X$ is compact and $f$ is continuous, then $Y$ is compact.

(c) If $X$ is connected, $f$ is continuous and $f(X)$ is dense in $Y$, then $Y$ is connected.

(d) If the set $\{(x, y) \in X \times Y: y=f(x)\}$ is closed in $X \times Y$ and $Y$ is compact, then $f$ is continuous.