Let $X$ be a topological space, equipped with its Borel $\sigma$-algebra $\mathcal B(X)$, and let $\mathbb P$ be a Radon probability measure on $(X, \mathcal B(X))$. Recall that the support of the measure $\mathbb P$ is the smallest closed set of full measure.

Is the support necessarily separable? If so, why? If not, what is a counterexample?