Show that if \(X\) is Lindelöf and \(Y\) is compact, then \(X \times Y\) is Lindelöf.
Solution
Goal: Prove that the product space \(X \times Y\) of a Lindelöf space \(X\) and a compact space \(Y\) is Lindelöf.
1. Slices \(\{x\} \times Y\) and finite subcovers:Let \(\mathcal{A}\) be any open cover of \(X \times Y\).For each \(x \in X\), there exists a finite subcollection \(\mathcal{A}_x \subset \mathcal{A}\) and an open neighborhood \(U_x \subseteq X\) of \(x\) such that:\[U_x \times Y \subseteq \bigcup_{A \in \mathcal{A}_x} A.\]Proof:
1.1. The slice \(\{x\} \times Y\) is homeomorphic to \(Y\), hence compact.
1.2. Since \(\mathcal{A}\) is an open cover of \(X \times Y\), it covers \(\{x\} \times Y\). By compactness of \(\{x\} \times Y\), there exists a finite subcollection \(\mathcal{A}_x = \{A_1, \dots, A_k\} \subset \mathcal{A}\) such that \(\{x\} \times Y \subseteq \bigcup_{i=1}^k A_i =: W_x\).
1.3. The set \(W_x\) is open in \(X \times Y\) and contains the slice \(\{x\} \times Y\).
1.4. Since \(Y\) is compact, the Tube Lemma applies: there exists an open neighborhood \(U_x\) of \(x\) in \(X\) such that \(U_x \times Y \subseteq W_x\).
2. Countable subcover of \(X\):There exists a countable sequence of points \((x_n)_{n=1}^\infty\) in \(X\) such that \(X = \bigcup_{n=1}^\infty U_{x_n}\).Proof: The collection \(\{U_x : x \in X\}\) is an open cover of \(X\). Since \(X\) is Lindelöf, there exists a countable subcover \(\{U_{x_n} : n \in \mathbb{Z}_+\}\).
3. Construction of the countable subcover of \(X \times Y\):The subcollection \(\mathcal{A}^* = \bigcup_{n=1}^\infty \mathcal{A}_{x_n} \subset \mathcal{A}\) is a countable open cover of \(X \times Y\).Proof:
3.1. Countability: Each \(\mathcal{A}_{x_n}\) is finite by
1. A countable union of finite sets is countable, so \(\mathcal{A}^*\) is countable.
1.2. Covering property: For any \((x, y) \in X \times Y\), 2 guarantees \(x \in U_{x_m}\) for some \(m \in \mathbb{Z}_+\).
1.3. Then \((x, y) \in U_{x_m} \times Y \subseteq \bigcup_{A \in \mathcal{A}_{x_m}} A \subseteq \bigcup_{A \in \mathcal{A}^*} A\).
1.4. Thus \(\mathcal{A}^*\) covers \(X \times Y\).
4. Conclusion:Every open cover \(\mathcal{A}\) of \(X \times Y\) admits a countable subcover \(\mathcal{A}^*\), so \(X \times Y\) is Lindelöf. Q.E.D.