Denote \(\displaystyle \overline{H}\) as the topological closure of H and \(\displaystyle \tau_{\ \overline{H}}\) as the subspace topology on \(\displaystyle \overline{H}\). Munkres claims in Ch2, exercise 3 that \(\displaystyle ( ( \overline{H} , \cdot ) , \tau_{ \ \overline{H}} )\) is a topological space and I'm trying to prove it. If I can prove \(\displaystyle \overline{H} \) is a subgroup of G, I can finish the proof (that the multiplication operator and inverse operators in \(\displaystyle \overline{H}\) are continuous).

All we have to work with is the structure induced on \(\displaystyle \overline{H}\) by the general topological group G. If this were a metric space, we could take convergent sequences and work with that. However, in a general topological space, we might work with topological convergence; if I take neighborhoods of a point of closure, I can get a sequence, but without compactness I don't see how to get a convergent sequence to the point of closure.

Ideas?