Consider the seventeen properties listed in Exercise 1 of the Supplementary Exercises (Review of the Basics).
Which of these properties are preserved under finite products? Countable products? Arbitrary products?
Solution
Goal: Classify the preservation behavior of the 17 fundamental topological properties under finite, countable, and arbitrary Cartesian products equipped with the product topology.
1. Complete classification table:- Preserved under arbitrary products (7):1. \(T_1\)2. Hausdorff (\(T_2\))3. Regular (\(T_3\))4. Completely regular (\(T_{3\frac{1}{2}}\))5. Compact (by Tychonoff’s Theorem)6. Connected7. Path-connected- Preserved under countable products, but NOT arbitrary (5):8. First-countable9. Second-countable10. Separable (by Hewitt-Marczewski-Pondiczery theorem up to cardinality \(\mathfrak{c}\))11. Metrizable12. Sequentially compact- Preserved under finite products, but NOT countable (2):13. Locally compact (an infinite product is locally compact \(\iff\) almost all factors are compact)14. Locally connected (an infinite product is locally connected \(\iff\) almost all factors are connected)- NOT preserved even under finite products (3):15. Normal (\(T_4\)) (e.g. Sorgenfrey plane \(\mathbb{R}_l \times \mathbb{R}_l\))16. Lindelöf (e.g. \(\mathbb{R}_l \times \mathbb{R}_l\))17. Limit point compact
2. Proof outlines for arbitrary products:Proof:
2.1. Separation axioms (\(T_1, T_2, T_3, T_{3\frac{1}{2}}\)): Points and coordinates can be separated in individual factor spaces; taking cylinder open sets \(\pi_\alpha^{-1}(U_\alpha)\) transfers separation to \(\prod X_\alpha\).
2.2. Compactness: Proved by Tychonoff’s Theorem.
2.3. Connectedness & Path-connectedness: A product is path-connected (resp. connected) by mapping coordinate paths \(\gamma_\alpha(t)\) (resp. using dense unions of finite cylinder sets).
3. Proof outlines and counterexamples for countable/finite/failing properties:Proof:
3.1. Countability and Metrizability (Countable YES, Arbitrary NO): A countable product of metric spaces is metrizable via \(D(\mathbf{x}, \mathbf{y}) = \sum_{n=1}^\infty 2^{-n} \bar{d}_n(x_n, y_n)\). Uncountable products such as \(\{0, 1\}^{\mathbb{R}}\) fail to be first-countable at any point.
3.2. Local Compactness & Local Connectedness (Finite YES, Infinite NO): For \(\mathbb{R}^\omega = \prod_{n=1}^\infty \mathbb{R}\), no basic open set has compact closure because basic open sets equal \(\mathbb{R}\) in all but finitely many coordinates.
3.3. Normality & Lindelöf (Finite NO): The Sorgenfrey line \(\mathbb{R}_l\) is normal and Lindelöf, but the Sorgenfrey plane \(\mathbb{R}_l \times \mathbb{R}_l\) contains the closed discrete anti-diagonal \(L = \{(x, -x)\}\), rendering it neither Lindelöf nor normal. Q.E.D.