Explain what is meant by a base for a topology. Illustrate your definition by describing bases for the topology induced by a metric on a set, and for the product topology on the cartesian product of two topological spaces.

A topological space $(X, \mathcal{T})$ is said to be separable if there is a countable subset $C \subseteq X$ which is dense, i.e. such that $C \cap U \neq \emptyset$ for every nonempty $U \in \mathcal{T}$. Show that a product of two separable spaces is separable. Show also that a metric space is separable if and only if its topology has a countable base, and deduce that every subspace of a separable metric space is separable.

Now let $X=\mathbb{R}$ with the topology $\mathcal{T}$ having as a base the set of all half-open intervals

$$[a, b)=\{x \in \mathbb{R}: a \leqslant x<b\}$$

with $a<b$. Show that $X$ is separable, but that the subspace $Y=\{(x,-x): x \in \mathbb{R}\}$ of $X \times X$ is not separable.

[You may assume standard results on countability.]