We know that if $G $ is a group and $A = \{ \tau_{i} \}_{i \in I}$ is a family of topological groups on group $G$ so that $\forall i \in I$ , $( G , \tau_{i} )$ is a topological group, and $\tau = \bigvee \tau_{i}$ ( sup $\tau_{i}$), then $( G , \tau)$ is a topological group.
Is it right to say:
withs mentioned conditions , $\bigwedge \tau_{i} = \tau $ ( inf $\tau_{i}$ ) is a topological group.
There is a counterexample: page 116 of this paper by Samuel has an example of two (!) group topologies on a set whose intersection (i.e. infimum) is not a group topology.