In Multidimensional diffusion processes by Stroock and Varadhan the probability measures $M(X)$ on a Polish space $X$ are considered as a subset of the dual of $C_b(X)$ (continuous bounded functions on $X$ with the sup-norm). Now $M(X)$ naturally inherits the weak* topology from $C_b(X)^*$.
The authors then proceed to give a metric that induces this topology. It can be shown (see note below) that $U_\rho(X)$ (uniformly continuous bounded functions on $X$ w.r.t $\rho$ with the sup-norm) is separable. Let $\left\{ \phi_k \right\}_{k \geq 0 }$ be dense in $U_\rho(X)$ and let $$\Delta(\mu, \lambda) = \sum \limits _ {k = 0}^\infty \frac{1}{2^{-k}} \frac{\left| \int_X \phi_k \, \mathrm{d}\mu - \int_X \phi_X \, \mathrm{d}\lambda \right|}{\left\| \phi_k \right\|}.$$ This is a metric on $M(X)$ that induces the weak* topology on $M(X)$.
A quote from the proof: "Obviously, the topology induced by $\Delta$ is weaker than the weak* topology." Why is this? I have tried to identify $B_\Delta(\mu,\varepsilon)$ as a preimage of an open set, but no luck so far.
Let $\Lambda_x: C_b(X)^* \to \mathbb{R}$ be such that $\Lambda_x f =f(x)$ with $x \in C_b(X)$ and $f \in C_b(X)^*$. The weak* topology is the topology given by these maps. Clearly $$ \Delta(\mu, \lambda) = \sum \limits _ {k = 0}^\infty \frac{1}{2^{-k}} \left| \Lambda_{\frac{\phi_k}{\left\|\phi_k \right\|}} (\mu - \lambda)\right|, $$ but I'm still unable to write the ball as a preimage of an open set.
Note: By Tychonoff's embedding theorem, $X$ is homeomorphic to a subset of a compact metric space and has an equivalent metric $\rho$ w.r.t which $X$ is totally bounded. The completion of $(X,\rho)$, denoted $\overline{X}$, is compact and hence $U_\rho(X)$ is isomorphic to $C(\overline{X})$ which is separable. (There is also a note that if two measures $\mu$ and $\lambda$ agree on $U_\rho(X)$ (as functionals), they are the same measure on the Borel $\sigma$-algebra of $X$.)