This might be an easy question, but I don't really know what to use.
Let $(S,\mathcal{S},d)$ be a seperable metric space, and let furthermore $(S,\mathcal{S})$ be standard borel
That is there exist a injective map so $\varphi:$ $(S,\mathcal{S} )\mapsto(\mathbb{R} ,\mathcal{B} (\mathbb{R} ))$ where $\varphi(S)\in \mathcal{B} (\mathbb{R} )$, $\varphi\in\mathcal{M}(\mathcal{S})$ and $\varphi^{-1}$ is $\mathcal{B} (\mathbb{R})-\mathcal{S}$-measurable.
Can I without further assumptions know my metric space contains all open (or closed) sets?