Different definitions of topological group

Question

Recently I discovered the definition of topological group. So, topological group is an abstract group $G$ endowed with topological structure such that the maps $mult: G\times G\longrightarrow G$ and $inv:G\longrightarrow G$ (multiplication and inversion) are continuous. I'd like to understand if it is enough just to suppose $mult$ to be continuous. Could you give me some slight hint?

Answer 1

No, it is not enough to assume that the multiplication map is continuous. See questions here and here.

However, if the topology on $G$ is nice enough, then it is enough to assume that just the multiplication map is continuous.

Theorem (Ellis, 1957): Let $G$ be a group with a locally compact Hausdorff topology such that the multiplication map $G \times G \rightarrow G$ is separately continuous. Then $G$ is a topological group.