Suppose is a topological group, i.e., a group with a topology and the multiplication and inverse operators are continuous. Let H be a subgroup of G. I can prove that H is a topological group. The problem is with .
Denote as the topological closure of H and as the subspace topology on . Munkres claims in Ch2, exercise 3 that is a topological space and I'm trying to prove it. If I can prove is a subgroup of G, I can finish the proof (that the multiplication operator and inverse operators in are continuous).
All we have to work with is the structure induced on 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.