To unite topology and group theory, we can define a topological group as a group $(G, \ast)$ with a topology such that $(x, y) \mapsto x \ast y$ and $x \mapsto x\inv$ are continuous functions. Only requiring the first to be continuous is not enough.
Some sources like Munkres additionally requires it to be $T_1$, which seemed arbitrary at first, why not Hausdorff instead? It turns out that (1) being Hausdorff (2) being $T_1$ (3) $\{e\}$ being a closed set; are all equivalent statements for topological groups. So one interesting way to prove some space is Hausdorff is to define a group operation on the space which is continuous and its inverse is continuous too, then prove that $\{e\}$ is closed. This is probably useless.