= Neighbourhood criterion for a topological group
= Neighborhood criterion for a topological group
{synonym}
A topology on a <group> makes it a <topological group> precisely when its <neighbourhood bases> satisfy two local tests: every neighbourhood $U$ of $xy$ contains $VW$ for some neighbourhoods $V$ of $x$ and $W$ of $y$, and every neighbourhood of $x^{-1}$ contains $V^{-1}$ for some neighbourhood $V$ of $x$. These are exactly continuity of multiplication in the <product topology> and continuity of inversion. Equivalently, bases can be chosen as left translates of a basis at the identity, with arbitrarily small squares, inverses and conjugates inside each prescribed identity neighbourhood.
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