A theorem of Katetov states that if is compact with a hereditarily normal cube , then is metrizable (discussed in this previous post). This means that for any non-metrizable compact space , Katetov’s theorem guarantees that some subspace of the cube is not normal. Where can a non-normal subspace of be found? Is it in , in or in ? In other words, what is the “dimension” in which the hereditary normality fails for a given compact non-metrizable (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.

____________________________________________________________________

**Katetov’s theorems**

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

*Theorem 1*

If is hereditarily normal (i.e. every one of its subspaces is normal), then one of the following condition holds:

- The factor is perfectly normal.
- Every countable and infinite subset of the factor is closed.

*Theorem 2*

If and are compact and is hereditarily normal, then both and are perfectly normal.

*Theorem 3*

Let be a compact space. If is hereditarily normal, then is metrizable.

____________________________________________________________________

**Examples of compact non-metrizable spaces**

The set-theoretic result presented here is usually motivated by looking at Theorem 3. The question is: Can in Theorem 3 be replaced by ? 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.

- One-point compactification of the Tychonoff plank.
- One-point compactification of where is a maximal almost disjoint family of subsets of .
- The first compact uncountable ordinal, i.e. .
- The one-point compactification of an uncountable discrete space.
- Alexandroff double circle.
- Double arrow space.
- 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, 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, is hereditarily normal. For #3 through #5, each has a closed subset that is not a set (hence not perfectly normal). In #3 and #4, the non--set is a single point. In #5, the the non--set is the inner circle. Thus the compact space 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 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:

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

*Question*Such a space would be an example to show that the condition “ 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 such that 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 such that 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 with hereditarily normal 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 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 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.

____________________________________________________________________

**Reference**

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

____________________________________________________________________