We say that a topological group $G$ is a $P$-group if every $G_\delta$ subset of $G$ is open, that is if the intersection of countably many open subsets is open.
According to a paper of Pestov it is evident that $P$-groups have a basis of neighbourhoods of the identity consisting of open subgroups, but that is not evident at all to me after thinking about it for a while. Why is this claim true?
Topological groups have the property that if $V\ni e$ is open, then there is an open $V'\ni e$ such that $V'\cdot V'\subseteq V$. You can also ensure that $V'$ is symmetric, by intersecting it with its inverse.
You can keep doing this inductively, and the intersection (after $\omega$ steps) will be a subgroup contained in $V$.
The hypothesis tells you that this subgroup is an open neighbourhood of $e$.