Suppose $\Omega:=C([0,1])$ is the space of continuous functions $\omega:[0,1]\to \mathbb R$. Let $S=(S_t)_{0\le t\le 1}$ be the coordinate process on $\Omega$, i.e.
$$S_t(\omega):=\omega(t),\quad \forall\;\; \omega\in \Omega\; \mbox{and}\; t \in [0,1].$$
Denote by $\mathcal P(\Omega)$ the set of probability measures on $\Omega$, and by $\mathcal P\subset\mathcal P(\Omega)$ the subset consisting of probability measures $\mathbb P$ s.t. $S$ is a square integrable martingale under $\mathbb P$ and it holds $\mathbb P-$almost surely that
$$dS_t = \sigma(t,S_t)dW^{\mathbb P}_t,\quad \forall t\in [0,1],$$
where $\sigma$ is some measurable function and $W^{\mathbb P}$ is a Wiener process under $\mathbb P$. Then, under which topology $\big(\mbox{on } \mathcal P(\Omega)\big)$ $\mathcal P$ can be closed?