Let $(X,\tau)$ be a topological space.

Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming form $\mathcal{E}$ and $\tau$ are the same. Can we conclude that $X$ is second countable ?!

Let $(X,\tau)$ be a topological space.

Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming form $\mathcal{E}$ and $\tau$ are the same. Can we conclude that $X$ is second countable ?!

This question is also asked when $X$ is a locally convex space. Please read the comments below.