When a product space is hereditarily normal

When the spaces X and Y are normal spaces, the product space X \times Y is not necessarily normal. Even if one of the factors is metrizable, there is still no guarantee that the product is normal. So it is possible that the normality of each of the factors X and Y can have no influence on the normality of the product X \times Y. The dynamics in the other direction are totally different. When the product X \times Y is hereditarily normal, the two factors X and Y are greatly impacted. In this post, we discuss a theorem of Katetov, which shows that the hereditary normality of the product can impose very strict conditions on the factors, which lead to many interesting results. This theorem also leads to an interesting set-theoretic result, and thus can possibly be a good entry point to the part of topology that deals with consistency and independence results – statements that cannot be proved true or false based on the generally accepted axioms of set theory (ZFC). In this post, we discuss Katetov’s theorem and its consequences. In the next post, we discuss examples that further motivate the set-theoretic angle.

A subset W of a space X is said to be a G_\delta-set in X if W is the intersection of countably many open subsets of X. A space X is perfectly normal if it is normal and that every closed subset of X is a G_\delta-set. Some authors use other statements to characterize perfect normality (here is one such characterization). Perfect normality implies hereditarily normal (see Theorem 6 in this previous post). The implication cannot be reversed. Katetov’s theorem implies that the hereditary normality of the product X \times Y will in many cases make one or both of the factors perfectly normal. Thus the hereditary normality in the product X \times Y is a very strong property.

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Katetov’s theorems

Theorem 1
If X \times Y is hereditarily normal (i.e. every one of its subspaces is normal), then one of the following condition holds:

  • The factor X is perfectly normal.
  • Every countable and infinite subset of the factor Y is closed.

Proof of Theorem 1
The strategy we use is to define a subspace of X \times Y that is not normal after assuming that none of the two conditions is true. So assume that X has a closed subspace W that is not a G_\delta-set and assume that T=\left\{t_n: n=1,2,3,\cdots \right\} is an infinite subset of Y that is not closed. Let p \in Y be a limit point of T such that p \notin T. The candidate for a non-normal subspace of X \times Y is:

    M=X \times Y-W \times \left\{p \right\}

Note that M is an open subspace of X \times Y since it is the result of subtracting a closed set from X \times Y. The following are the two closed sets that demonstrate that M is not normal.

    H=W \times (Y-\left\{p \right\})

    K=(X-W) \times \left\{p \right\}

It is clear that H and K are closed subsets of M. Let U and V be open subsets of M such that H \subset U and K \subset V. We show that U \cap V \ne \varnothing. To this end, define U_j=\left\{x \in X: (x,t_j) \in U \right\} for each j. It follows that for each j, W \subset U_j. Furthermore each U_j is an open subspace of X. Thus W \subset \bigcap_j U_j. Since W is not a G_\delta-set in X, there must exist t \in \bigcap_j U_j such that t \notin W. Then (t, p) \in K and (t, p) \in V.

Since V is open in the product X \times Y, choose open sets A \subset X and B \subset Y such that (t,p) \in A \times B and A \times B \subset V. With p \in B, there exists some j such that t_j \in B. First, (t,t_j) \in V. Since t \in U_j, (t,t_j) \in U. Thus U \cap V \ne \varnothing. This completes the proof that the subspace M is not normal and that X \times Y is not hereditarily normal. \blacksquare

Let’s see what happens in Theorem 1 when both factors are compact. If both X and Y are compact and if X \times Y is hereditarily normal, then both X and Y must be perfect normal. Note that in any infinite compact space, not every countably infinite subset is closed. Thus if compact spaces satisfy the conclusion of Theorem 1, they must be perfectly normal. Hence we have the following theorem.

Theorem 2
If X and Y are compact and X \times Y is hereditarily normal, then both X and Y are perfectly normal.

Moe interestingly, Theorem 1 leads to a metrization theorem for compact spaces.

Theorem 3
Let X be a compact space. If X^3=X \times X \times X is hereditarily normal, then X is metrizable.

Proof of Theorem 3
Suppose that X^3 is hereditarily normal. By Theorem 2, the compact spaces X^2 and X are perfectly normal. In particular, the following subset of X^2 is a G_\delta-set in X^2.

    \Delta=\left\{(x,x): x \in X \right\}

The set \Delta is said to be the diagonal of the space X. It is a well known result that any compact space whose diagonal is a G_\delta-set in the square is metrizable (discussed here). \blacksquare

The results discussed here make it clear that hereditary normality in product spaces is a very strong property. One obvious question is whether Theorem 3 can be improved by assuming only the hereditary normality of X^2. This was indeed posted by Katetov himself. This leads to the discussion in the next post.

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Reference

  1. Engelking R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.

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\copyright \ 2015 \text{ by Dan Ma}

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One thought on “When a product space is hereditarily normal

  1. Pingback: Normality in Cp(X) | Dan Ma's Topology Blog

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