 Neighbourhood criterion for a topological group

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.