What are some applications of "separable" spaces?

Question

A separable space is a space that contains a countable dense subset.

For example, the space of continuous functions $C[a,b]$ is separable.

Are there some practical applications arising out of this fact?

Answer 1

Consider a sequence of quadrature rules $Q_n$ for integrating continuous functions $f$. You have

$$|I(f)-Q_n(f)| \leq |I(f)-I(p)|+|I(p)-Q_n(p)|+|Q_n(p)-Q_n(f)|$$

where $I$ is the integration functional and $p$ is an arbitrary function. The bound is useful if $p$ is close to $f$.

Weierstrass's theorem (from which separability of $C[a,b]$ is usually proven) implies that you can make the error in the first term small by taking $p$ to be a suitable polynomial. This reduces the problem to controlling $|I(p)-Q_n(p)|$ (i.e. ensuring that the method is accurate for polynomials) and $|Q_n(p)-Q_n(f)|$ (i.e. ensuring a kind of "stability" for $Q_n$ itself). The details for these two terms depend on the rule you're talking about.

For another application, suppose you have a sequence of functions $f_n$ defined on a space $A$, whose ranges are all contained in some fixed sequentially compact set $K$. If $C$ is a countable subset of $A$, then you can use diagonalization to extract a subsequence which converges pointwise on $C$. If $C$ was dense and $f_n$ was "nice", then there is some hope of extending the convergence from the countable set to the entire space. There are several results of this type; the Arzela-Ascoli theorem is probably the best-known example.