Let $(X,\tau)$ be a topological vector space. Suppose that, there is a sequence of subsets $X_n\subseteq X$ with,

For every $n\in \mathbb{N}$, the topology $\tau$ on $X_n$ is second countable and metrizable space.

$X_n\subseteq X_{n+1}$ and $X=\bigcup X_n$.

Q. Is the Borel sigma algebra coming from the weak topology $\sigma(X,X^*)$ is the same as the Borel sigma algebra coning from $\tau$?