(a) For a subset $A$ of a topological space $X$, define the closure cl $(A)$ of $A$. Let $f: X \rightarrow Y$ be a map to a topological space $Y$. Prove that $f$ is continuous if and only if $f(c l(A)) \subseteq c l(f(A))$, for each $A \subseteq X$.

(b) Let $M$ be a metric space. A subset $S$ of $M$ is called dense in $M$ if the closure of $S$ is equal to $M$.

Prove that if a metric space $M$ is compact then it has a countable subset which is dense in $M$.