Consider the seventeen properties listed in Exercise 1 of the Supplementary Exercises (Review of the Basics).
Which of these properties does a compact Hausdorff space have?
Solution
Goal: Evaluate which of the 17 fundamental topological properties are necessarily possessed by every compact Hausdorff space, and provide proofs or counterexamples for each.
1. The seventeen standard properties and their validity for compact Hausdorff spaces:- Guaranteed properties (9):1. \(T_1\) (Points are closed)2. Hausdorff (\(T_2\))3. Regular (\(T_3\))4. Completely regular (\(T_{3\frac{1}{2}}\))5. Normal (\(T_4\))6. Compact7. Limit point compact8. Locally compact9. Lindelöf- Not necessarily possessed (8):10. First-countable11. Second-countable12. Separable13. Metrizable14. Sequentially compact15. Connected16. Path-connected17. Locally connected
2. Proofs for guaranteed properties:Proof:
2.1. \(T_1\) and Hausdorff: Every Hausdorff space is \(T_1\), and Hausdorffness holds by hypothesis.
2.2. Regularity and Normality: Every compact Hausdorff space is normal (\(T_4\)). Given disjoint closed sets \(A, B \subseteq X\), compactness allows finite subcovers of point-separating open sets to separate \(A\) and \(B\). Normality with \(T_1\) immediately implies regularity (\(T_3\)).
2.3. Complete regularity: By Urysohn’s Lemma, every normal \(T_1\) space is completely regular (\(T_{3\frac{1}{2}}\)).
2.4. Compactness, Limit Point Compactness, Local Compactness, Lindelöf:- Compactness holds by hypothesis.- Compactness implies limit point compactness (every infinite subset has an accumulation point).- Every compact space is locally compact (\(X\) itself serves as a compact neighborhood for each point).- Every open cover has a finite subcover, which is in particular a countable subcover, so \(X\) is Lindelöf.
3. Counterexamples for non-guaranteed properties:Proof:
3.1. Countability and Metrizability (First-countable, Second-countable, Separable, Metrizable):- The one-point compactification of an uncountable discrete space is compact Hausdorff, but fails to be first-countable (and hence not second-countable, separable, or metrizable).- The product space \(\{0, 1\}^{\mathbb{R}}\) is compact Hausdorff by Tychonoff’s Theorem, but is not first-countable and not metrizable.
3.2. Sequential compactness: The space \(\{0, 1\}^{\mathbb{R}}\) (or the Stone-Čech compactification \(\beta\mathbb{N}\)) is compact Hausdorff but not sequentially compact.
3.3. Connectedness properties (Connected, Path-connected, Locally connected):- The two-point discrete space \(\{0, 1\}\) or the Cantor set \(C\) is compact Hausdorff, but is totally disconnected (hence neither connected, path-connected, nor locally connected without isolated components). Q.E.D.