Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?

clearly every closed neighborhood of $b$ contains $a$. So I wonder if any open neighborhood of $1$ contains a closed neighborhood of $1$.

If the answer to the above question is negative I add another assumption:

How if we have a neighborhood $U$ of $1$, with $\overline{U^{-1}}$ compact?