Is the closure of a subgroup of a quasitopological group a subgroup?

Question

The closure of a subgroup in a topological group is a subgroup. Is the same true for a quasitopological group?

Answer 1

Just to have it as an answer, my comment from MO:

Yes: if $h_n \to g$ and $h'_m \to g'$, then $h_n h'_m \to g h'_m$ for each $m$, so each $g h'_m$ lies in the closure; and then $g h'_m \to g g'$, so $g g'$ lies in the closure. (Identity and inverses are obvious.)