Give the definition of a metric on a set $X$ and explain how this defines a topology on $X$.

Suppose $(X, d)$ is a metric space and $U$ is an open set in $X$. Let $x, y \in X$ and $\epsilon>0$ such that the open ball $B_{\epsilon}(y) \subseteq U$ and $x \in B_{\epsilon / 2}(y)$. Prove that $y \in B_{\epsilon / 2}(x) \subseteq U$.

Explain what it means (i) for a set $S$ to be dense in $X$, (ii) to say $\mathcal{B}$ is a base for a topology $\mathcal{T}$.

Prove that any metric space which contains a countable dense set has a countable basis.