# weak convergence of probability measures on a topological but non-metrizable space

Let $X$ be a topological space and let $\mathcal{B}$ be the Borel $\sigma$-algebra. Let $\Delta$ be the space of all (countably-additive) probability measures on $(X,\mathcal{B})$. Can I define on $\Delta$ the topology of weak convergence as this topology is typically defined when $X$ is metrizable? I imagine Portmanteau's lemma will not go through unless $X$ has further structure, but the definition of weak convergence should not require more than a topology on $X$. Is this correct?