This post discusses the property of having a -diagonal and related diagonal properties. The focus is on the diagonal properties in between -diagonal and submetrizability. The discussion is followed by a diagram displaying the relative strengths of these properties. Some examples and questions are discussed.

**G-delta Diagonal**

In any space , a subset is said to be a -set in the space (or is a -subset of ) if is the intersection of countably many open subsets of . A subset of is an -set in (or is an -subset of ) if is the union of countably closed subsets of the space . Of course, the set is a -set if and only if , the complement of , is an -set.

The diagonal of the space is the set , which is a subset of the square . When the set is a -set in the space , we say that the space has a -diagonal.

It is straightforward to verify that the space is a Hausdorff space if and only if the diagonal is a closed subset of . As a result, if is a Hausdorff space such that is perfectly normal, then the diagonal would be a closed set and thus a -set. Such spaces, including metric spaces, would have a -diagonal. Thus any metric space has a -diagonal.

A space is submetrizable if there is a metrizable topology that is weaker than the topology for . Then the diagonal would be a -set with respect to the weaker metrizable topology of and thus with respect to the orginal topology of . This means that the class of spaces having -diagonals also include the submetrizable spaces. As a result, Sorgenfrey line and Michael line have -diagonals since the Euclidean topology are weaker than both topologies.

A space having a -diagonal is a simple topological property. Such spaces form a wide class of spaces containing many familiar spaces. According to the authors in [2], the property of having a -diagonal is an important ingredient of submetrizability and metrizability. For example, any compact space with a -diagonal is metrizable (see this blog post). Any paracompact or Lindelof space with a -diagonal is submetrizable. Spaces with -diagonals are also interesting in their own right. It is a property that had been research extensively. It is also a current research topic; see [7].

**A Closer Look**

To make the discussion more interesting, let’s point out a few essential definitions and notations. Let be a space. Let be a collection of subsets of . Let . The notation refers to the set . In other words, is the union of all the sets in that intersect the set . The set is also called the star of the set with respect to the collection .

If , we write instead of . Then refers to the union of all sets in that contain the point . The set is then called the star of the point with respect to the collection .

Note that the statement of having a -diagonal is defined by a statement about the product . It is desirable to have a translation that is a statement about the space .

*Theorem 1*

Let be a space. Then the following statements are equivalent.

- The space has a -diagonal.
- There exists a sequence of open covers of such that for each , .

The sequence of open covers in condition 2 is called a -diagonal sequence for the space . According to condition 2, at any given point, the stars of the point with respect to the open covers in the sequence collapse to the given point.

One advantage of a -diagonal sequence is that it is entirely about points of the space . Thus we can work with such sequences of open covers of instead of the -set in . Theorem 1 is not a word for word translation. However, the proof is quote natural.

Suppose that where each is an open subset of . Then let . It can be verify that is a -diagonal sequence for .

Suppose that is a -diagonal sequence for . For each , let . It follows that .

It is informative to compare the property of -diagonal with the definition of Moore spaces. A development for the space is a sequence of open covers of such that for each , is a local base at the point . A space is said to be developable if it has a development. The space is said to be a Moore space if is a Hausdorff and regular space that has a development.

The stars of a given point with respect to the open covers of a development form a local base at the given point, and thus collapse to the given point. Thus a development is also a -diagonal sequence. It then follows that any Moore space has a -diagonal.

A point in a space is a -point if the point is the intersection of countably many open sets. Then having a -diagonal sequence implies that that every point of the space is a -point since every point is the intersection of the stars of that point with respect to a -diagonal sequence. In contrast, any Moore space is necessarily a first countable space since the stars of any given point with respect to the development is a countable local base at the given point. The parallel suggests that spaces with -diagonals can be thought of as a weak form of Moore spaces (at least a weak form of developable spaces).

**Regular G-delta Diagonal**

We discuss other diagonal properties. The space is said to have a regular -diagonal if where each is an open subset of such that . This diagonal property also has an equivalent condition in terms of a diagonal sequence.

*Theorem 2*

Let be a space. Then the following statements are equivalent.

- The space has a regular -diagonal.
- There exists a sequence of open covers of such that for every two distinct points , there exist open sets and with and and there also exists an such that no member of intersects both and .

For convenience, we call the sequence described in Theorem 2 a regular -diagonal sequence. It is clear that if the diagonal of a space is a regular -diagonal, then it is a -diagonal. It can also be verified that a regular -diagonal sequence is also a -diagonal sequence. To see this, let be a regular -diagonal sequence for . Suppose that and . Choose open sets and and an integer guaranteed by the regular -diagonal sequence. Since , choose such that . Then would be an element of that meets both and , a contradiction. Then for all .

To proof Theorem 2, suppose that has a regular -diagonal. Let where each is open in and . For each , let be the collection of all open subsets of such that . It can be verified that is a regular -diagonal sequence for .

On the other hand, suppose that is a regular -diagonal sequence for . For each , let . It can be verified that .

**Rank-k Diagonals**

Metric spaces and submetrizable spaces have regular -diagonals. We discuss this fact after introducing another set of diagonal properties. First some notations. For any family of subsets of the space and for any , define . For any integer , let . Thus is the star of the star with respect to and is the star of and so on.

Let be a space. A sequence of open covers of is said to be a rank- diagonal sequence of if for each , we have . When the space has a rank- diagonal sequence, the space is said to have a rank- diagonal. Clearly a rank-1 diagonal sequence is simply a -diagonal sequence as defined in Theorem 1. Thus having a rank-1 diagonal is the same as having a -diagonal.

It is also clear that having a higher rank diagonal implies having a lower rank diagonal. This follows from the fact that a rank diagonal sequence is also a rank diagonal sequence.

The following lemma builds intuition of the rank- diagonal sequence. For any two distinct points and of a space , and for any integer , a -link path from to is a set of open sets such that , and for all . By default, a single open set containing both and is a d-link path from to for any integer .

*Lemma 3*

Let be a space. Let be a positive integer. Let be a sequence of open covers of . Then the following statements are equivalent.

- The sequence is a rank- diagonal sequence for the space .
- For any two distinct points and of , there is an integer such that .
- For any two distinct points and of , there is an integer such that there is no -link path from to consisting of elements of .

It can be seen directly from definition that Condition 1 and Condition 2 are equivalent. For Condition 3, observe that the set is the union of types of open sets – open sets in containing , open sets in that intersect the first type, open sets in that intersect the second type and so on down to the open sets in that intersect . A path is formed by taking one open set from each type.

We now show a few basic results that provide further insight on the rank- diagonal.

*Theorem 4*

Let be a space. If the space has a rank-2 diagonal, then is a Hausdorff space.

*Theorem 5*

Let be a Moore space. Then has a rank-2 diagonal.

*Theorem 6*

Let be a space. If has a rank-3 diagonal, then has a regular -diagonal.

Once Lemma 3 is understood, Theorem 4 is also easily understood. If a space has a rank-2 diagonal sequence , then for any two distinct points and , we can always find an where there is no 2-link path from to . Then and can be separated by open sets in . Thus these diagonal ranking properties confer separation axioms. We usually start off a topology discussion by assuming a reasonable separation axiom (usually implicitly). The fact that the diagonal ranking gives a bonus makes it even more interesting. Apparently many authors agree since -diagonal and related topics had been researched extensively over decades.

To prove Theorem 5, let be a development for the space . Let and be two distinct points of . We claim that there exists some such that . Suppose not. This means that for each , . This also means that for each . Choose for each . Since is a Moore space, is a local base at . Then converges to . Since is a local base at , converges to , a contradiction. Thus the claim that there exists some such that is true. By Lemma 3, a development for a Moore space is a rank-2 diagonal sequence.

To prove Theorem 6, let be a rank-3 diagonal sequence for the space . We show that is also a regular -diagonal sequence for . Suppose and are two distinct points of . By Lemma 3, there exists an such that there is no 3-link path consisting of open sets in that goes from to . Choose with . Choose with . Then it follows that no member of can intersect both and (otherwise there would be a 3-link path from to ). Thus is also a regular -diagonal sequence for .

We now show that metric spaces have rank- diagonal for all integer .

*Theorem 7*

Let be a metrizable space. Then has rank- diagonal for all integers .

If is a metric that generates the topology of , and if is the collection of all open subsets with diameters with respect to the metrix then is a rank- diagonal sequence for for any integer .

We instead prove Theorem 7 topologically. To this end, we use an appropriate metrization theorem. The following theorem is a good candidate.

** Alexandrov-Urysohn Metrization Theorem**. A space is metrizable if and only if the space has a development such that for any with , the set is contained in some element of . See Theorem 1.5 in p. 427 of [5].

Let be the development from Alexandrov-Urysohn Metrization Theorem. It is a development with a strong property. Each open cover in the development refines the preceding open cover in a special way. This refinement property allows us to show that it is a rank- diagonal sequence for for any integer .

First, we make a few observations about . From the statement of the theorem, each is a refinement of . As a result of this observation, is a refinement of for any . Furthermore, for each , for any .

Let with . Based on the preceding observations, it follows that there exists some such that . We claim that there exists some integer such that there are no -link path from to consisting of open sets from . Then is a rank- diagonal sequence for according to Lemma 3.

We show this claim is true for . Observe that there cannot exist such that , and . If there exists such a pair, then would be contained in and , a contradiction. Putting it in another way, there cannot be any 2-link path from to such that the open sets in the path are from . According to Lemma 3, the sequence is a rank-2 diagonal sequence for the space .

In general for any , there cannot exist any -link path from to such that the open sets in the path are from . The argument goes just like the one for the case for . Suppose the path exists. Using the special property of , the 2-link path is contained in some open set in . The path is now contained in a -link path consisting of elements from the open cover . Continuing the refinement process, the path is contained in a 2-link path from to consisting of elements from . Like before this would lead to a contradiction. According to Lemma 3, is a rank- diagonal sequence for the space for any integer .

Of course, any metric space already has a -diagonal. We conclude that any metrizable space has a rank- diagonal for any integer .

We have the following corollary.

*Corollary 8*

Let be a submetrizable space. Then has rank- diagonal for all integer .

In a submetrizable space, the weaker metrizable topology has a rank- diagonal sequence, which in turn is a rank- diagonal sequence in the original topology.

**Examples and Questions**

The preceding discussion focuses on properties that are in between -diagonal and submetrizability. In fact, one of the properties has infinitely many levels (rank- diagonal for integers ). We would like to have a diagram showing the relative strengths of these properties. Before we do so, consider one more diagonal property.

Let be a space. The set is said to be a zero-set in if there is a continuous such that . In other words, a zero-set is a set that is the inverse image of zero for some continuous real-valued function defined on the space in question.

A space has a zero-set diagonal if the diagonal is a zero-set in . The space having a zero-set diagonal implies that has a regular -diagonal, and thus a -diagonal. To see this, suppose that where is continuous. Then where . Thus having a zero-set diagonal is a strong property.

We have the following diagram.

The diagram summarizes the preceding discussion. From top to bottom, the stronger properties are at the top. From left to right, the stronger properties are on the left. The diagram shows several properties in between -diagonal at the bottom and submetrizability at the top.

Note that the statement at the very bottom is not explicitly a diagonal property. It is placed at the bottom because of the classic result that any compact space with a -diagonal is metrizable.

In the diagram, “rank-k diagonal” means that the space has a rank- diagonal where is an integer, which in terms means that the space has a rank- diagonal sequence as defined above. Thus rank- diagonal is not to be confused with the rank of a diagonal. The rank of the diagonal of a given space is the largest integer such that the space has a rank- diagonal. For example, for a space that has a rank-2 diagonal but has no rank-3 diagonal, the rank of the diagonal is 2.

To further make sense of the diagram, let’s examine examples.

The Mrowka space is a classic example of a space with a -diagonal that is not submetrizable (introduced here). Where is this space located in the diagram? The Mrowka space, also called Psi-space, is defined using a maximal almost disjoint family of subsets of . We denote such a space by where is a maximal almost disjoint family of subsets of . It is a pseudocompact Moore space that is not submetrizable. As a Moore space, it has a rank-2 diagonal sequence. A well known result states that any pseudocompact space with a regular -diagonal is metrizable (see here). As a non-submetrizable space, the Mrowka space cannot have a regular -diagonal. Thus is an example of a space with a rank-2 diagonal but not a rank-3 diagonal sequence.

Examples of non-submetrizable spaces with stronger diagonal properties are harder to come by. We discuss examples that are found in the literature.

Example 2.9 in [2] is a Tychonoff separable Moore space that has a rank-3 diagonal but not of higher diagonal rank. As a result of not having a rank-4 diagonal, is not submetrizable. Thus is an example of a space with rank-3 diagonal (hence with a regular -diagonal) that is not submetrizable. According to a result in [6], any separable space with a zero-set diagonal is submetrizable. Then the space is an example of a space with a regular -diagonal that does not have a zero-set diagonal. In fact, the authors of [2] indicated that this is the first such example.

Example 2.9 of [2] shows that having a rank-3 diagonal does not imply having a zero-set diagonal. If a space is strengthened to have a rank-4 diagonal, does it imply having a zero-set diagonal? This is essentially Problem 2.13 in [2].

On the other hand, having a rank-3 diagonal implies a rank-2 diagonal. If we weaken the hypothesis to just having a regular regular -diagonal, does it imply having a rank-2 diagonal? This is essentially Problem 2.14 in [2].

The authors of [2] conjectured that for each , there exists a space with a rank- diagonal but not having a rank- diagonal. This conjecture was answered affirmatively in [8] by constructing, for each integer , a Tychonoff space with a rank- diagonal but not having a rank- diagonal. Thus even for high , a non-submetrizable space can be found with rank- diagonal.

One natural question is this. Is there a non-submetrizable space that has rank- diagonal for all ? We have not seen this question stated in the literature. But it is clearly a natural question.

Example 2.17 in [2] is a non-submetrizable Moore space that has a zero-set diagonal and has rank-3 diagonal exactly (i.e. it does not have a higher rank diagonal). This example shows that having a zero-set diagonal does not imply having a rank-4 diagonal. A natural question is then this. Does having a zero-set diagonal imply having a rank-3 diagonal? This appears to be an open question. This is hinted by Problem 2.19 in [2]. It asks, if is a normal space with a zero-set diagonal, does have at least a rank-2 diagonal?

The property of having a -diagonal and related properties is a topic that had been researched extensively over the decades. It is still an active topic of research. The discussion in this post only touches on the surface. There are many other diagonal properties not covered here. To further investigate, check with the papers listed below and also consult with information available in the literature.

**Reference**

- Arhangelskii A. V., Burke D. K.,
*Spaces with a regular -diagonal*, Topology and its Applications, Vol. 153, No. 11, 1917–1929, 2006. - Arhangelskii A. V., Buzyakova R. Z.,
*The rank of the diagonal and submetrizability*, Comment. Math. Univ. Carolinae, Vol. 47, No. 4, 585-597, 2006. - Buzyakova R. Z.,
*Cardinalities of ccc-spaces with regular -diagonals*, Topology and its Applications, Vol. 153, 1696–1698, 2006. - Buzyakova R. Z.,
*Observations on spaces with zeroset or regular -diagonals*, Comment. Math. Univ. Carolinae, Vol. 46, No. 3, 469-473, 2005. - Gruenhage, G.,
*Generalized Metric Spaces*, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 423-501, 1984. - Martin H. W.,
*Contractibility of topological spaces onto metric spaces*, Pacific J. Math., Vol. 61, No. 1, 209-217, 1975. - Xuan Wei-Feng, Shi Wei-Xue,
*On spaces with rank k-diagonals or zeroset diagonals*, Topology Proceddings, Vol. 51, 245{251, 2018. - Yu Zuoming, Yun Ziqiu,
*A note on the rank of diagonals*, Topology and its Applications, Vol. 157, 1011–1014, 2010.

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