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.