= Solution
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> is
$$
\begin{aligned}&U\in\mathcal B_{xy}\ \Longrightarrow\ \exists V\in\mathcal B_x,\ W\in\mathcal B_y:\ VW\subseteq U,\\&U\in\mathcal B_{x^{-1}}\ \Longrightarrow\ \exists V\in\mathcal B_x:\ V^{-1}\subseteq U.\end{aligned}
$$
Necessity 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 have
$$
\boxed{\exists V\in\mathcal B_e:\ V^2\subseteq U;\qquad\exists V\in\mathcal B_e:\ V^{-1}\subseteq U;\qquad\forall g\in G\ \exists V\in\mathcal B_e:\ gVg^{-1}\subseteq U.}
$$
These 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 $xHx^{-1}\subseteq U$, then $V$ with $V^{-1}\subseteq H$; thus $(xV)^{-1}=x^{-1}(xV^{-1}x^{-1})\subseteq x^{-1}U$. 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.
Back to article page
