This is a small attempt toward making sense of the spaces with small diagonal. What makes the property of small diagonal interesting is the longstanding open problem that is discussed below. We use examples to help make sense of the definition of small diagonal. Then we discuss briefly the open problem regarding small diagonal. We discuss the following three examples of compact spaces:
- with the ordered topology
- The one-point compactification of a discrete space of cardinality
- The double arrow space
By a space we mean a topological space that is Tychonoff, i.e. Hausdorff and completely regular (defined here). Let be a space. The subset of the square is called the diagonal of the space .
We want to focus on two diagonal properties. The space is said to have a -diagonal if is a -set in , i.e. is the intersection of countably many open subsets of . The space is said to have a small diagonal if for each uncountable subset of , there is an open subset of such that and misses uncountably many points of .
How do these two diagonal properties relate? Any space that has a -diagonal also has a small diagonal. This fact can be worked out quite easily based on the definitions. The opposite direction is a totally different matter. In fact, the question of whether having a small diagonal implies having a -diagonal is related to a longstanding open question.
Let’s focus on compact spaces. A classic metrization theorem for compact spaces states that any compact space with a -diagonal is metrizable (see here). A natural question is: if a compact space has a small diagonal, must it be metrizable? Indeed this is a well known open problem, a longstanding problem that has not completely resolved completely. Before discussion the open problem, we look at the three examples indicated above.
The three examples are all non-metrizable compact spaces and thus do not have a -diagonal. We show that they also not have a small diagonal. To appreciate the definition of small diagonal, it is helpful to look spaces that do not have a small diagonal.
Example 1
Consider with the order topology. Using interval notion, . This is the space of all countable ordinals plus the point at the end. This is a compact space that is not metrizable. For example, any compact metrizable space would be separable. The last point cannot be in the closure of any countable subset. According to the classic theorem mentioned above, cannot have a -diagonal. We show that it does have a small diagonal too.
Let . Note that any open set of the form contains the point and contains all but countably many points of . As a result, any open set containing the diagonal contains all but countably many points of . This violates the definition of having a small diagonal. Thus the space does not have a small diagonal.
More on the Definition
Example 1 suggests a different angle in looking at the definition. The set is a -length sequence convergent to the diagonal, meaning that any open set containing the diagonal contains a tail of the sequence (in this case containing all but countably many elements in the sequence). The convergent sequence view point is the way Husek defined small diagonal [7].
In [7], is said to have an -accessible diagonal if there is an -length sequence that converges to the diagonal , meaning that any open set containing contains all but countably many terms in the sequence. This is exactly what is occurring in the space . The set is precisely a convergent sequence converging to the diagonal. The space has an -accessible diagonal.
Seen in this light, spaces with a small diagonal are the the spaces that do not have an -accessible diagonal (or spaces that have an -inaccessible diagonal). Though the definition of Husek is more descriptive, the term small diagonal, suggested by E. van Douwen, has become more popular. The small diagonal defined above is more positive sounding. For example, we say the space has a small diagonal. In Husek [7], the spaces of interest would be the spaces without an -accessible diagonal. This definition is a negative one (defined by the lack of certain thing) and takes more syllables to express. Personally speaking, we prefer the term small diagonal though we understand that -accessible diagonal is more descriptive.
A slightly different (but equivalent) way of stating the definition of a space with a small diagonal: a space has a small diagonal if for any uncountable , there is an uncountable such that . In the literature, the definition of a space with a small diagonal is usually either this one or the one given at the beginning of this post.
Example 2
Now consider the space where points in are isolated and open neighborhoods of the point are of the form with being any finite subset of . This is usually called the one-point compactification of a discrete space (in this case of size ). This space is compact. It is also non-metrizable since it has an uncountable discrete subset. Thus it cannot have a -diagonal. The weight of here is . Any compact space whose weight is cannot have a small diagonal. This is Fact 1 below.
Some Basic Results
Fact 1
Let any compact space with . Then does not have a small diagonal.
Proof of Fact 1
Let be a base for . Let , which is a base for . Define . Since is compact, the diagonal is compact. As a result, . Furthermore, .
We claim that is a base for the diagonal . To show this, let be an open subset of such that . We can assume that is the union of elements of the base . Let for some . Since is compact, for some finite . Note that and that .
Enumerate as . Since is compact and has uncountable weight, is not metrizable. Hence the diagonal is not a -set. As a result, for each , . Furthermore, for any countable , .
Pick . For any with , choose .
Then the sequence converges to the diagonal . To see this, fix . From the way the sequence is chosen, for all . This concludes the proof that does not have a small diagonal.
Fact 2
Let any compact space with . Then if has a small diagonal, then and thus is metrizable.
Fact 2 is an easy corollary of Fact 1. Fact 2 says that any compact space with “small” weight (no more than ) is metrizable if it has a small diagonal. Both Example 1 and Example 2 are compact non-metrizable spaces with small weight. Therefore they do not have small diagonal. The following basic fact is also useful.
Fact 3
Let any compact space. Then if has a small diagonal, then , i.e. is countably tight.
Proof of Fact 3
Suppose is uncountably tight. We show that it does not have a small diagonal. A sequence of points is a free sequence of length if for each , . For any compact space with tightness at least , there exists a free sequence of length (see Lemma 2 here). Thus in the space in question, there exists a free sequence .
The main result in [9] says that if a compact space contains a free sequence of length , then it contains a free sequence of the same length that is convergent. We now assume that the above free sequence is also convergent, i.e. it converges to some point . Thus every open set containing contains all but countably many . Consider the sequence . Observe that every open set in containing the point contains all but countably pairs . This implies that every open set containing the diagonal contains all but countably many points in the sequence. This shows that the compact does not have a small diagonal.
According to Fact 3, the compact ordinal does not have a small diagonal since it is uncountably tight at the last point .
Example 3
We now consider the double arrow space. Let . The space consists of two copies of the unit interval, an upper one and a lower one. See the first two diagrams in this previous post. For , a basic open set containing the point in the upper interval is of the form . For , a basic open set containing the point in the lower interval is of the form . The rightmost point in the upper interval and the leftmost point in the lower interval are made isolated points.
The double arrow space is compact, perfectly normal and not metrizable (discussed here). Thus does not have a -diagonal. We do not have a direct way of showing that it does not have a small diagonal. We rely on a result from [5], which says that every compact metrizably fibered space with a small diagonal is metrizable. In light of this result, we only need to show that the double arrow space is metrizably fibered. A space is metrizably fibered if there is a continuous map from onto some metrizable space such that each point inverse is metrizable.
Starting with the double arrow space , let be defined by for each . This is a two-to-one continuous map from the double arrow space onto the unit interval. The map is essentially a quotient map, the result of identifying as one point . By the result in [5], the double arrow space cannot have a small diagonal.
The Open Problem
As mentioned earlier, what makes the property of small diagonal is a related longstanding open problem. The statement “every compact space with a -diagonal is metrizable” is true in ZFC. What about the following statement?
(*) Every compact space with a small diagonal is metrizable.
This question whether this statement is true was raised in Husek [7]. As of the writing of this article, the problem is still unsolved. There are partial results, some consistent results and some ZFC results.
Assuming CH, the answer to Hesek’s question is positive. Husek [7] showed that under CH, every compact space with a small diagonal such that the tightness of is countable is metrizable. Fact 3 from above states that every compact space with a small diagonal has countble tightness. Combining the two results, it follows that under CH any compact space with a small diagonal is metrizable. Fact 3 followed from a result by Juhasz and Szentmikloss [9]. Dow and Pavlov [3] showed that under PFA, every compact space with a small diagonal is metrizable.
There are partial answers in ZFC. We mention three results. The first one is Fact 2 discussed above, which is that compact spaces with small diagonal are metrizable if there is a weight restriction (weight no more than ). If there exists a non-metrizable compact space with a small diagonal, its weight would have to be greater than .
Gruenhage [5] showed that every compact metrizably fibered space with a small diagonal is metrizable (this result is discussed in Example 3 above). Dow and Hart [2] showed that every compact space with a small diagonal that is weight fibered is metrizable. The notion of weight fibered is a generalization of metrizably fibered. A space is weight fibered if there is a continuous surjection such that and each point inverse have weight at most .
The statement “every countably compact space with a -diagonal is metrizable” is a true statement in ZFC (see here). How about the statement “every countably compact space with a small diagonal is metrizable”, the statement (*) above with compact replaced by countably compact? Gruenhage [5] showed that this statement is consistent with and independent of ZFC. To prove or disprove this statement extra set theory assumptions beyond ZFC are required. This provides an interesting contrast between compact and countably compact with respect to the open problem. By broadening Husek’s question from compact to countably compact, the statement cannot be settled in ZFC. Dow and Pavlov [3] provided two consistent examples of “countably compact non-metrizable with small diagonal” that are improvements upon examples from Gruenhage.
Now consider the statement “Lindelof space with a small diagonal must have a -diagonal.” Dow and Pavlov [3] provided a consistent negative answer, an example of a Lindelof space with a small diagonal that does not have a -diagonal (under negation of CH).
Another broad natural question is: what do compact spaces with small diagonal look like? The main problem is, of course, trying to see if these spaces are metrizable. There have been attempts to explore this general question of what these spaces look like. According to Juhasz and Szentmikloss [9], these spaces have countable tightness (Fact 2 above). However, it is not known if compact spaces with small diagonal have points of countable character. Dow and Hart [2] uncovered a surprising connection that if there is a subset of the real line that is a Luzin set, every compact space with a small diagonal does have points of countable character. Dow and Hart in the same paper also showed that in every compact space with a small diagonal, CCC subspaces have countable -weight.
This is a brief walk through of the open problem based on the statement (*) indicated above. To find out more, consult with the references listed below. Beyond the main open problem, there are many angles to be explored.
Reference
- Arhangelskii, A., Bella, A., Few observations on topological spaces with small diagonal, Zb. Rad. Filoz. Fak. Nisu, 6, No. 2, 211-213, 1992.
- Dow, A., Hart, P. Elementary chains and compact spaces with a small diagonal, Indagationes Mathematicae, 23, No. 3, 438-447, 2012.
- Dow, A., Pavlov, O. More about spaces with a small diagonal, Fund. Math., 191, No. 1, 67-80, 2006.
- Dow, A., Pavlov, O. Perfect preimages and small diagonal, Topology Proc., 31, No. 1, 89-95, 2007.
- Gruenhage, G., Spaces having a small diagonal, Topology Appl., 22, 183-200, 2002.
- Gruenhage, G., Generalized metrizable spaces, Recent Progress in General Topology III (K.P. Hart, J. van Mill, and P. Simon, eds.), Atlantis Press 2014.
- Husek, M., Topological spaces without -accessible diagonal, Comment. Math. Univ. Carolin., 18, No. 4, 777-788, 1977.
- Juhasz, I., Cardinals Functions II, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 63-109, 1984.
- Juhasz, I., Szentmikloss, Z. Convergent free sequences in compact spaces, Proc. Amer. Math. Soc., 116, No. 4, 1153-1160, 1992.
- Zhou, H. X., On The Small Diagonals, Topology Appl., 13, 283-293, 1982.
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