Solution
= Solution
If $a\notin A$, every point of $A$ is isolated by (ii). The set $A$ is the union of these open singletons, so $\boxed{A\text{ is open in }(X,d_a)}$, including the empty case.
= Solution
If $a\notin A$, every point of $A$ is isolated by (ii). The set $A$ is the union of these open singletons, so $\boxed{A\text{ is open in }(X,d_a)}$, including the empty case.