Compactness vs. Sequential Compactness

COMPACTNESS VS. SEQUENTIAL COMPACTNESS The aim of this handout is to provide a detailed proof of the equivalence between the two definitions of compactness: existence of a finite subcover of any open cover, and existence of a limit point of any infinite subset. Definition 1. K is compact if every open cover of K contains a finite subcover. K is sequentially compact if every infinite subset of K has a limit point in K. Theorem 1. K is compact ⇐⇒ K is sequentially compact. The first half of this statement (compact =⇒ sequentially compact) is Theorem 2.37 in Rudin and is proved there. Our aim is to prove the converse implication (sequentially compact =⇒ compact), following the lines of Exercises 23, 24 and 26 in Rudin Chapter 2. The proof requires the introduction of two auxiliary notions: Definition 2. A space X is separable if it admits a countable dense subset. For example R is separable (Q is countable, and it is dense since every real number is a limit of rationals); for the same reason Rk is separable (consider all points with only rational coordinates). Definition 3. A collection {Vα } of open subsets of X is said to be a base for X if the following is true: for every x ∈ X and for every open set G ⊂ X such that x ∈ G, there exists α such that x ∈ Vα ⊂ G. In other words, every open subset of X decomposes as a union of a subcollection of the Vα ’s – the Vα ’s “generate” all open subsets. The family {Vα } almost always contains infinitely many members (the only exception is if X is finite). However, if X happens to be separable, then countably many open subsets are enough to form a base (the converse statement is also true and is an easy exercise): Lemma 1. Every separable metric space has a countable base. Proof. Assume X is separable: by definition it contains a countable dense subset P = {p1 , p2 , . . . }. Consider the countable collection of neighborhoods {Nr (pi ), r ∈ Q, i = 1, 2, . . . }. We show that it is a base by checking the definition. Consider any open set G ⊂ X and any point x ∈ G. Since G is open, there exists r > 0 such that Nr (x) ⊂ G. Decreasing r if necessary we can assume without loss of generality that r is rational. Since P is dense, by definition x is a limit point of P , so Nr/2 (x) contains a point of P . So there exists i such that d(x, pi ) < 2r . Since r is rational, the neighborhood Nr/2 (pi ) belongs to the chosen collection. Moreover, Nr/2 (pi ) ⊂ Nr (x) ⊂ G. Finally, since d(x, pi ) < 2r we also have x ∈ Nr/2 (pi ). So the chosen collection is a base for X. � Lemma 2. If X is sequentially compact then it is separable. Proof. Fix δ > 0, and let x1 ∈ X. Choose x2 ∈ X such that d(x1 , x2 ) ≥ δ, if possible. Having chosen x1 , . . . , xj , choose xj+1 (if possible) such that d(xi , xj+1 ) ≥ δ for all i = 1, . . . , j. We first notice that this process has to stop after a finite number of iterations. Indeed, otherwise we would obtain an infinite sequence of points xi mutually distant by at least δ; since X is sequentially compact the infinite subset {xi , i = 1, 2, . . . } would admit a limit point y, and the neighborhood Nδ/2 (y) would contain infinitely many of the xi ’s, contradicting the fact that any two of them are distant by at least δ. So after a finite number of iterations we obtain x1 , . . . , xj such that Nδ (x1 )∪. . . Nδ (xj ) = X (every point of X is at distance < δ from one of the xi ’s). We now consider this construction for δ = n1 (n = 1, 2, . . . ). For n = 1 the construction gives points x11 , . . . , x1j1 such that N1 (x11 ) ∪ · · · ∪ N1 (x1j1 ) = X, for n = 2 we get x21 , . . . , x2j2 such that N1/2 (x21 ∪ · · · ∪ N1/2 (x2j2 ) = X, and so on. Let S = {xki , k ≥ 1, 1 ≤ i ≤ jk }: clearly S is countable. We claim that S is dense (i.e. S̄ = X). Indeed, if x ∈ X and r > 0, the neighborhood Nr (x) always contains at least a point of S (choosing n so that n1 < r, one of the xni ’s is at distance less than r � from x), so every point of X either belongs to S or is a limit point of S, i.e. S̄ = X. At this point we know that every sequentially compact set has a countable base. We now show that this is enough to get countable subcovers of any open cover. Lemma 3. If X has a countable base, then every open cover of X admits an at most countable subcover. Proof. Homework � The final ingredient is the following: Lemma 4. If {Fn } is a sequence of non-empty closed subsets of a sequentially compact set K such � that Fn ⊃ Fn+1 for all n = 1, 2, . . . , then ∞ F 6 ∅. n=1 n = (Since we know at this point that every compact set is sequentially compact, and since compact subsets are closed, this lemma implies immediately the Corollary to Theorem 2.36 in Rudin). Proof. Take xn ∈ Fn for each integer n, and let E = {xn , n = 1, 2, . . . }. If E is finite then one of the xi belongs to infinitely many Fn ’s. Since F1 ⊃ F2 ⊃ . . . , this implies that xi belongs to every � Fn , and we get that ∞ F n=1 n is not empty. Assume now that E is infinite. Since K is sequentially compact, E has a limit point y. Fix a value of n: every neighborhood of y contains infinitely many points of E; among them, we can find one which is of the form xi for i ≥ n and therefore belongs to Fn (because xi ∈ Fi ⊂ Fn ). Since every neighborhood of y contains a point of Fn , we get that either y ∈ Fn , or y is a limit point of Fn ; however since Fn is closed, every limit point of Fn belongs to F�n . So in either case we conclude that y ∈ Fn . Since this holds for every n, we obtain that y ∈ ∞ n=1 Fn , which proves that the intersection is not empty. � We can now prove the theorem. Assume that K is sequentially compact, and let {Gα } be an open cover of K. By Lemma 1 and Lemma 2, K has a countable base, so by Lemma 3 {Gα } admits an at most countable subcover that we will denote {Gi }i≥1 . Our aim is to show that {Gi } admits a finite subcover (which will also be a finite subcover of {Gα }). If {Gi } only contains finitely many members, we are already done; so assume that there are infinitely many Gi ’s, and assume that for every value of n we have G1 ∪ · · · ∪ Gn 6⊃ K (else we have found a finite subcover). Let Fn = {x ∈ K, x 6∈ G1 ∪ · · · ∪ Gn } = K ∩ Gc1 ∩ · · · ∩ Gcn . Because the Gi are open, Fn is closed; by assumption �∞Fn is non-empty; and clearly Fn ⊃ Fn+1 for all n. Therefore, applying Lemma 4 we obtain that n=1 Fn = 6 ∅, i.e. there exists a point y ∈ K such that y 6∈ G1 ∪ · · · ∪ Gn for every n. We conclude that y 6∈ ∞ i=1 Gi , which is a contradiction since the open sets Gi cover K. So there exists a value of n such that G1 , . . . , Gn cover K. We conclude that every open cover of K admits a finite subcover, and therefore that K is compact.