A Corollary from Munkres' Topology

Question

This exercise comes from Munkres' Topology, 2nd edition, Page 188. It says

11. Corollary. Let $G$ be a topological group; let $A$ and $B$ be subsets of $G$. If $A$ is closed in $G$ and $B$ is compact, then $A\cdot B$ is closed in $G$.[ Hint: First give a proof using sequences, assuming that $G$ is metrizable. ]

This seems to be nontrivial to me. I know there is a proof without using nets, but I'm fascinated by the way with nets. I have been pondering this exercise all the whole day and cannot figure out how to use nets to prove it. Currently I learn topology on my own so I did some research, but it still resists my efforts to solve it. There is a hint but I cannot benefit from it. I need more hints, or even a proof using nets. Any help will be appreciated. Thanks.

Answer 1

A proof using nets is straightforward enough: suppose $f: I \to G$ is a net from a directed set $(I,\le_I)$ to $G$ such that all $f(i) \in A\cdot B$ and suppose that $f \to p \in G$. We need to show that $p \in A \cdot B$ as well (a set is closed iff it's closed under limits of nets).

We start by writing $f(i) = a_i \cdot b_i$ for some $a_i \in A, b_i \in B$ for each $i \in I$. Then $f_B: I \to G$ defined by $f_B(i) = b_i$ is a net with values in the compact set $B$ so by a characterisation of compactness using nets, there is a subnet $g_B: J \to G$ of $f_B$ and some $b_0 \in B$ such that $g_B \to b_0$, and where $(J, \le_J)$ is some directed set and there is a connection function $c: J \to I$ witnessing the subnet relation.

First we conclude that the net $j \to f(c(j))g_B(j)^{-1}$ converges to $p \cdot b_0^{-1}$ by continuity of the group operation. As all $f(c(j))g_B(j)^{-1} \in A$ we conclude by closedness of $A$ that $p\cdot b_0^{-1} \in A$, so that $p = (p \cdot b_0^{-1})\cdot b_0 \in A \cdot B$, as required.

Note that it uses the exercises 6,7 and 10 before it. (This is exercise 11). It's a nice exercise to show the naturalness of net-proofs, IMHO.

Answer 2

This is the proof for sequences: We want to show that the limit of any convergent sequence in $A\cdot B$ is in $A\cdot B$.

So if $a_nb_n\in A\cdot B$ and $a_nb_n\to g\in G$, then we want to show $g\in A\cdot B$. Choose a subsequence such that $b_{n_k}\to b\in B$ (using compactness). Then $a_{n_k}=a_{n_k}b_{n_k}b_{n_k}^{-1}\to gb^{-1}$ so (as $A$ is closed) $gb^{-1}\in A$. Hence $g=gb^{-1}b\in A\cdot B$. So $A\cdot B$ is closed.

Now can you rewrite this with nets instead of sequences?