Looking for non-normal subspaces of the square of a compact X

A theorem of Katetov states that if X is compact with a hereditarily normal cube X^3, then X is metrizable (discussed in this previous post). This means that for any non-metrizable compact space X, Katetov’s theorem guarantees that some subspace of the cube X^3 is not normal. Where can a non-normal subspace of X^3 be found? Is it in X, in X^2 or in X^3? In other words, what is the “dimension” in which the hereditary normality fails for a given compact non-metrizable X (1, 2 or 3)? Katetov’s theorem guarantees that the dimension must be at most 3. Out of curiosity, we gather a few compact non-metrizable spaces. They are discussed below. In this post, we motivate an independence result using these examples.

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Katetov’s theorems

First we state the results of Katetov for reference. These results are proved in this previous post.

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.

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

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

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Examples of compact non-metrizable spaces

The set-theoretic result presented here is usually motivated by looking at Theorem 3. The question is: Can X^3 in Theorem 3 be replaced by X^2? We take a different angle of looking at some standard compact non-metric spaces and arrive at the same result. The following is a small listing of compact non-metrizable spaces. Each example in this list is defined in ZFC alone, i.e. no additional axioms are used beyond the generally accepted axioms of set theory.

  1. One-point compactification of the Tychonoff plank.
  2. One-point compactification of \psi(\mathcal{A}) where \mathcal{A} is a maximal almost disjoint family of subsets of \omega.
  3. The first compact uncountable ordinal, i.e. \omega_1+1.
  4. The one-point compactification of an uncountable discrete space.
  5. Alexandroff double circle.
  6. Double arrow space.
  7. Unit square with the lexicographic order.

Since each example in the list is compact and non-metrizable, the cube of each space must not be hereditarily normal according to Theorem 3 above. Where does the hereditary normality fail? For #1 and #2, X is a compactification of a non-normal space and thus not hereditarily normal. So the dimension for the failure of hereditary normality is 1 for #1 and #2.

For #3 through #7, X is hereditarily normal. For #3 through #5, each X has a closed subset that is not a G_\delta set (hence not perfectly normal). In #3 and #4, the non-G_\delta-set is a single point. In #5, the the non-G_\delta-set is the inner circle. Thus the compact space X in #3 through #5 is not perfectly normal. By Theorem 2, the dimension for the failure of hereditary normality is 2 for #3 through #5.

For #6 and #7, each X^2 contains a copy of the Sorgenfrey plane. Thus the dimension for the failure of hereditary normality is also 2 for #6 and #7.

In the small sample of compact non-metrizable spaces just highlighted, the failure of hereditary normality occurs in “dimension” 1 or 2. Naturally, one can ask:

    Question. Is there an example of a compact non-metrizable space X such that the failure of hereditary nornmality occurs in “dimension” 3? Specifically, is there a compact non-metrizable X such that X^2 is hereditarily normal but X^3 is not hereditarily normal?

Such a space X would be an example to show that the condition “X^3 is hereditarily normal” in Theorem 3 is necessary. In other words, the hypothesis in Theorem 3 cannot be weakened if the example just described were to exist.

The above list of compact non-metrizable spaces is a small one. They are fairly standard examples for compact non-metrizable spaces. Could there be some esoteric example out there that fits the description? It turns out that there are such examples. In [1], Gruenhage and Nyikos constructed a compact non-metrizable X such that X^2 is hereditarily normal. The construction was done using MA + not CH (Martin’s Axiom coupled with the negation of the continuum hypothesis). In that same paper, they also constructed another another example using CH. With the examples from [1], one immediate question was whether the additional set-theoretic axioms of MA + not CH (or CH) was necessary. Could a compact non-metrizable X such that X^2 is hereditarily normal be still constructed without using any axioms beyond ZFC, the generally accepted axioms of set theory? For a relatively short period of time, this was an open question.

In 2001, Larson and Todorcevic [3] showed that it is consistent with ZFC that every compact X with hereditarily normal X^2 is metrizable. In other words, there is a model of set theory that is consistent with ZFC in which Theorem 3 can be improved to assuming X^2 is hereditarily normal. Thus it is impossible to settle the above question without assuming additional axioms beyond those of ZFC. This means that if a compact non-metrizable X is constructed without using any axiom beyond ZFC (such as those in the small list above), the hereditary normality must fail at dimension 1 or 2. Numerous other examples can be added to the above small list. Looking at these ZFC examples can help us appreciate the results in [1] and [3]. These ZFC examples are excellent training ground for general topology.

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Reference

  1. Gruenhage G., Nyikos P. J., Normality in X^2 for Compact X, Trans. Amer. Math. Soc., Vol 340, No 2 (1993), 563-586
  2. Katetov M., Complete normality of Cartesian products, Fund. Math., 35 (1948), 271-274
  3. Larson P., Todorcevic S., KATETOV’S PROBLEM, Trans. Amer. Math. Soc., Vol 354, No 5 (2001), 1783-1791

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\copyright \ 2015 \text{ by Dan Ma}

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