I see a result on a textbook but I do not know how to prove it. let $(X,d)$ be a separable metric space and $A \subset X$. Suppose that the topology induced on $A$ by the metric $d_A$ coincides with the topology induced on $A$ by the topology on $X$ by the metric $d$.Then the metric space $(A,d_A)$ is separable.
Of course we have to find a countable dense set of points in $A$. But I cannot find way to do this because although we know there exists a countable dense subset of $X$, some points in $A$ may be approximated by a sequence of points in $X-A$, and I don't know how to find points in $A$ to replace them.
Can anyone help?
A metric space is separable if and only if it is second countable, and a subspace of a second countable space is second countable. Therefore, ...