Show that if \(G\) is a locally compact topological group and \(H\) is a subgroup, then \(G/H\) is locally compact.
Solution
1. Let \(\pi : G \to G/H\) be the quotient map, and let \(gH \in G/H\) be an arbitrary coset.
fix a point of \(G/H\).
2. Since \(G\) is locally compact, there is a compact neighborhood \(K\) of \(g\) in \(G\).
definition of local compactness.
3. \(\pi(K)\) is a compact neighborhood of \(gH\) in \(G/H\).
\(\pi\) is continuous and surjective, so \(\pi(K)\) is compact; and \(\pi\) is an open map (the quotient map of a topological group by a subgroup is open), so \(\pi(K)\) is a neighborhood of \(\pi(g) = gH\).
4. Hence every point of \(G/H\) has a compact neighborhood, so \(G/H\) is locally compact.
1 and 3.
5. Q.E.D.
4.