When all powers of a space are normal

It follows from the Tychonoff Theorem that if a topological space X is a compact Hausdorff space, then the product space X^\tau is normal for any cardinal number \tau. The converse is also true. If X^\tau is normal for any cardinal number \tau, then X must be compact. This is a theorem that is due to Noble (see [5]). The proof in [5] is a corollary resulted from a long chain of previous results of Noble and others. Many authors had produced simpler and more direct proofs of Noble’s theorem (e.g. [2], [3] and [4]). All these more direct proofs make use of the fact that the product space \omega^{\omega_1} is not normal (due to A. H. Stone). All of them except [2] make use of other strong topological results in order to derive Noble’s theorem. In [2], Engelking established Noble’s theorem by an elementary proof. In this post, we present the proof in [2] in full details. Noble’s theorem is also given in Engelking’s textbook as an exercise (see 3.12.15 on p. 233 in [1]).

Before proceeding to the main theorem, let’s set up some notation for working with the product space \prod_{\alpha \in S} X_\alpha. For x \in \prod_{\alpha \in S} X_\alpha, the \alpha^{th} coordinate of x is denoted by x_\alpha or (x)_\alpha. For M \subset S, the map P_M: \prod_{\alpha \in S} X_\alpha \rightarrow \prod_{\alpha \in M} X_\alpha is the natural projection map. In the product space \prod_{\alpha \in S} X_\alpha, standard basic open sets are of the form \prod_{\alpha \in S} O_\alpha where O_\alpha= X_\alpha for all but finitely many \alpha. We use supp(\prod_{\alpha \in S} O_\alpha) to denote the set of the finite number of \alpha \in S where O_\alpha \ne X_\alpha.

Noble’s Theorem

    If each power of a space X is normal, then X is compact.

Proof
Suppose that X^\tau is normal for all cardinal numbers \tau. Suppose that X is not compact. Then there exists a collection \mathcal{H}=\left\{H_\alpha: \alpha \in S \right\} of closed subsets of X such that \mathcal{H} has the finite intersection property but has empty intersection. Let H=\prod_{\alpha \in S} H_\alpha, which is a subspace of the product space \prod_{\alpha \in S} X_\alpha where each X_\alpha=X. We can also denote the product space \prod_{\alpha \in S} X_\alpha by X^\tau where \tau=\lvert S \lvert.

Let K=\left\{k \in X^{\lvert S \lvert}: \forall \ \beta, \gamma \in S, \ k_\beta=k_\gamma \right\}. Note that K is commonly referred to as the diagonal of the product space in question. Both H and K are closed sets in the product space X^\tau. Because the collection \mathcal{H} has empty intersection, H and K are disjoint closed sets. Since X^\tau is normal, there exist disjoint open subsets U and V of X^\tau such that H \subset U and K \subset V.

Let x_1 \in H. Let O_1 be a basic standard open set with x \in O_1 \subset U. Let S_1=supp(O_1). Then we have P^{-1}_{S_1}(P_{S_1}(x_1)) \subset U. Since \mathcal{H} has the finite intersection property, choose a_1 \in \bigcap_{\alpha \in S_1} H_\alpha. Then define x_2 \in H such that (x_2)_\alpha=a_1 for all \alpha \in S_1 and (x_2)_\alpha=(x_1)_\alpha for all \alpha \in S-S_1.

Let O_2 be a basic standard open set with x_2 \in O_2 \subset U. Let S_2=supp(O_2). By making O_2 a smaller open set if necessary, we can have S_1 \subset S_2. Then we have P^{-1}_{S_2}(P_{S_2}(x_2)) \subset U. Choose a_2 \in \bigcap_{\alpha \in S_2} H_\alpha. Then define x_3 \in H such that (x_3)_\alpha=a_2 for all \alpha \in S_2 and (x_3)_\alpha=(x_2)_\alpha for all \alpha \in S-S_2.

After this inductive process is completed, we can obtain:

  • a sequence x_1,x_2,x_3,\cdots of points of H=\prod_{\alpha \in S} H_\alpha,
  • a sequence S_1 \subset S_2 \subset S_3, \subset \cdots of finite subsets of the index set S,
  • a sequence a_1,a_2,a_3,\cdots of points of X

such that for each n \ge 2, (x_n)_\alpha=a_{n-1} for all \alpha \in S_{n-1} and P^{-1}_{S_n}(P_{S_n}(x_n)) \subset U.

By A. H. Stone’s theorem (Theorem 5 in [6]), X cannot contain a closed copy of \mathbb{N} (the space of the positive integers with the discrete topology). A proof that \mathbb{N}^{\omega_1} is also found in this post. Let A=\left\{a_1,a_2,a_3,\cdots \right\}. Either A is infinite or finite.

Case 1
Assume that A is an infinite set. Then A has a limit point a, meaning that every open subset of X containing a contains some a_n different from a. For each n \ge 2, define y_n \in \prod_{\alpha \in S} X_\alpha such that

  • (y_n)_\alpha=(x_n)_\alpha=a_{n-1} for all \alpha \in S_n
  • (y_n)_\alpha=a for all \alpha \in S-S_n

It is the case that y_n \in P^{-1}_{S_n}(P_{S_n}(x_n)) \subset U for all n, since y_n agrees with x_n on the finite set S_n. Let t \in K such that t_\alpha=a for all \alpha \in S. It follows that t is a limit point of \left\{y_2,y_3,y_4,\cdots \right\}. Thus t \in \overline{U}. Since t \in K \subset V, the open set V would have to contain points of U. But U and V are supposed to be disjoint open subsets of the product space \prod_{\alpha \in S} X_\alpha. Thus we have a contradiction.

Case 2
Assume that A is a finite set. Then for some m, a_j=a_m for all j \ge m. For each n \ge 2, define y_n \in \prod_{\alpha \in S} X_\alpha such that

  • (y_n)_\alpha=(x_n)_\alpha=a_{n-1} for all \alpha \in S_n
  • (y_n)_\alpha=a_m for all \alpha \in S-S_n

Let t \in K such that t_\alpha=a_m for all \alpha \in S. Then y_n=t for all n \ge m+1. As in Case 1, y_n \in P^{-1}_{S_n}(P_{S_n}(x_n)) \subset U for all n, implying that t \in K \cap U. This is a contradiction, since K \subset V and U and V are supposed to be disjoint.

Both cases lead to a contradiction. Thus if all powers of X is normal, X must be compact. This completes the proof of the theorem.

____________________________________________________________________

Reference

  1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  2. Engelking, R., An elementary proof of Noble’s theorem on normality of powers, Comment. Math. Univ. Carolinae, 29.4, 677-678, 1988.
  3. Franklin, S. P., Walker, R. C., Normalit of powers implies compactness, Proc. Amer. Math. Soc., 36, 295-296, 1972.
  4. Keesling, J., Normality and infinite product spaces, Adv. in. Math., 9, 90-92, 1972.
  5. Noble, N., Products with closed projections, II, Trans. Amer. Math. Soc., 160, 169-183, 1971.
  6. Ross, K. A., Stone, A. H. Products of separable spaces, Amer. Math. Monthly, 71, 398-403, 1964.

____________________________________________________________________

\copyright \ 2014 \text{ by Dan Ma}

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s