Let \(G\) be a topological group.
Let \(A\) and \(B\) be subspaces of \(G\). If \(A\) is closed and \(B\) is compact, show \(A \cdot B\) is closed. [Hint: If \(c\) is not in \(A \cdot B\), find a neighborhood \(W\) of \(c\) such that \(W \cdot B^{-1}\) is disjoint from \(A\).]
Let \(H\) be a subgroup of \(G\); let \(p: G \to G/H\) be the quotient map. If \(H\) is compact, show that \(p\) is a closed map.
Let \(H\) be a compact subgroup of \(G\). Show that if \(G/H\) is compact, then \(G\) is compact.
Solution
Goal: Prove that the product of a closed set and a compact set in a topological group is closed, deduce that quotient maps by compact subgroups are closed, and prove that group extensions by compact groups are compact.
1. Part (a): If \(A\) is closed and \(B\) is compact in a topological group \(G\), then \(A \cdot B\) is closed.Proof:
1.1. We show that \(G \setminus (A \cdot B)\) is open. Let \(c \in G \setminus (A \cdot B)\).
1.2. Then for every \(b \in B\), \(c b^{-1} \notin A\), so \(c b^{-1} \in G \setminus A\).
1.3. Since \(A\) is closed, \(G \setminus A\) is open.
1.4. The map \(\phi: G \times G \to G\) defined by \(\phi(x, y) = x y^{-1}\) is continuous.
1.5. For each \(b \in B\), since \(\phi(c, b) = c b^{-1} \in G \setminus A\), continuity of \(\phi\) provides open neighborhoods \(U_b\) of \(c\) and \(V_b\) of \(b\) such that \(U_b \cdot V_b^{-1} \subseteq G \setminus A\).
1.6. The open sets \(\{V_b : b \in B\}\) cover the compact set \(B\). Choose a finite subcover \(V_{b_1}, \dots, V_{b_k}\) of \(B\).
1.7. Define \(W = \bigcap_{i=1}^k U_{b_i}\). As a finite intersection of open neighborhoods of \(c\), \(W\) is an open neighborhood of \(c\).
1.8. For any \(w \in W\) and \(b \in B\), \(b \in V_{b_i}\) for some \(i\), so \(w b^{-1} \in U_{b_i} \cdot V_{b_i}^{-1} \subseteq G \setminus A\), meaning \(w b^{-1} \notin A\).
1.9. Thus \(w \notin A \cdot B\) for all \(w \in W\), so \(W \cap (A \cdot B) = \emptyset\).
1.10. Hence \(W \subseteq G \setminus (A \cdot B)\), proving that \(G \setminus (A \cdot B)\) is open and \(A \cdot B\) is closed.
2. Part (b): If \(H \le G\) is compact, then the quotient map \(p: G \to G/H\) is a closed map.Proof:
2.1. Let \(C \subseteq G\) be closed.
2.2. Under the quotient topology on \(G/H\), \(p(C)\) is closed in \(G/H\) if and only if \(p^{-1}(p(C))\) is closed in \(G\).
2.3. We identify \(p^{-1}(p(C))\):\[p^{-1}(p(C)) = \{g \in G : gH \in p(C)\} = \{g \in G : gH = cH \text{ for some } c \in C\} = C \cdot H.\]
2.4. Since \(C\) is closed and \(H\) is compact, by 1 the product \(C \cdot H\) is closed in \(G\).
2.5. Thus \(p^{-1}(p(C))\) is closed in \(G\), so \(p(C)\) is closed in \(G/H\). Hence \(p\) is a closed map.
3. Part (c): If \(H \le G\) is compact and \(G/H\) is compact, then \(G\) is compact.Proof:
3.1. Let \(\mathcal{U} = \{O_\alpha\}_{\alpha \in I}\) be an open cover of \(G\).
3.2. For each \(x \in G\), the coset \(xH = L_x(H)\) is compact as the continuous image of the compact set \(H\) under the left translation homeomorphism \(L_x\).
3.3. Since \(xH\) is compact, there exists a finite subset \(I_x \subset I\) such that \(xH \subseteq V_x := \bigcup_{\alpha \in I_x} O_\alpha\).
3.4. The set \(V_x\) is open in \(G\), so \(C_x = G \setminus V_x\) is closed in \(G\).
3.5. By 2, \(p\) is a closed map, so \(p(C_x)\) is closed in \(G/H\), and \(W_x = (G/H) \setminus p(C_x)\) is open in \(G/H\).
3.6. Since \(xH \cap C_x = \emptyset\), \(xH \notin p(C_x)\), so \(xH \in W_x\).
3.7. If \(yH \in W_x\), then \(yH \cap C_x = \emptyset\), so \(yH \subseteq V_x = \bigcup_{\alpha \in I_x} O_\alpha\).
3.8. The family \(\{W_x : xH \in G/H\}\) is an open cover of \(G/H\).
3.9. Since \(G/H\) is compact, there exist finitely many points \(x_1, \dots, x_m \in G\) such that \(G/H = \bigcup_{j=1}^m W_{x_j}\).
3.10. Therefore \(G = \bigcup_{j=1}^m p^{-1}(W_{x_j}) \subseteq \bigcup_{j=1}^m V_{x_j} = \bigcup_{j=1}^m \bigcup_{\alpha \in I_{x_j}} O_\alpha\).
3.11. This is a finite subcover of \(\mathcal{U}\) covering \(G\). Hence \(G\) is compact. Q.E.D.