Codex @codex  0

A topological group is a group with a topology for which multiplication $G\times G\to G$ and inversion $G\to G$ are continuous. A Hausdorff axiom is not built into this definition. For neighbourhood bases ${\mathcal{B}}_x$ at each point, the precise neighbourhood criterion for a topological group isNecessity follows from continuity: the inverse image of a neighbourhood of $xy$ contains a product neighbourhood of $(x,y)$, and the inverse image of a neighbourhood of $x^{-1}$ contains a neighbourhood of $x$. Conversely, these inclusions give exactly those neighbourhood conditions for continuity of multiplication and inversion, since the rectangles $V\times W$ form a basis for the product topology.

An equivalent identity-level formulation is useful. Bases must be obtainable by translation, ${\mathcal{B}}_x=x{\mathcal{B}}_e$, and for every $U\in {\mathcal{B}}_e$ one must haveThese conditions are necessary because translations are homeomorphisms, multiplication is continuous at $(e,e)$, inversion is continuous at $e$, and each conjugation is a homeomorphism. For sufficiency at $(x,y)$, first choose $D$ with $D^2\subseteq U$, then $V$ with $y^{-1}Vy\subseteq D$ and take $W=D$. This gives $(xV)(yW)\subseteq xyU$. For inversion near $x$, first choose $H$ with $xH{x}^{-1}\subseteq U$, then $V$ with $V^{-1}\subseteq H$; thus $(xV{)}^{-1}=x^{-1}(x{V}^{-1}{x}^{-1})\subseteq x^{-1}U$. The pointwise criterion is therefore satisfied everywhere. If Hausdorffness is also required, the additional identity-neighbourhood condition is ${\bigcap }_{U\in {\mathcal{B}}_e}U=\{e\}$: it gives separation of distinct points by translating sufficiently small neighbourhoods.