Are the following statements true or false? Give a proof or a counterexample as appropriate.

(i) If $f: X \rightarrow Y$ is a continuous map of topological spaces and $S \subseteq X$ is compact then $f(S)$ is compact.

(ii) If $f: X \rightarrow Y$ is a continuous map of topological spaces and $K \subseteq Y$ is compact then $\left.f^{-1}(K)=\{x \in X: f(x) \in K\}\right\}$ is compact.

(iii) If a metric space $M$ is complete and a metric space $T$ is homeomorphic to $M$ then $T$ is complete.