topological group in finer topology

Question

If $(G, \tau)$ is a topological group and $\tau \subseteq \tau_{1}$, is
$(G, \tau_{1})$ a topological group? What if $\tau_{1} \subseteq \tau$?

Answer 1

Let $G=(\mathbb R,+).$

Let $\tau_0$ be the usual topology on $\mathbb R.$

Let $\tau_1$ be the Sorgenfrey topology; basic open sets are half-open intervals of the form $[a,b).$

Let $\tau_2$ be the discrete topology on $\mathbb R.$

Then $\tau_0\subseteq\tau_1\subseteq\tau_2,$ and $(G,\tau_0)$ and $(G,\tau_2)$ are topological groups, but $(G,\tau_1)$ is not a topological group.