Two posts (the previous post and this post) are devoted to discussing the behavior of countable tightness in taking Cartesian products. The previous post shows that countable tightness behaves well in the product operation if the spaces are compact. In this post, we step away from the orderly setting of compact spaces. We examine the behavior of countable tightness in product of sequential fans. In this post, we show that countable tightness can easily be destroyed when taking products of sequential fans. Due to the combinatorial nature of sequential fans, the problem of determining the tightness of products of fans is often times a set-theoretic problem. In many instances, it is hard to determine the tightness of a product of two sequential fans without using extra set theory axioms beyond ZFC. The sequential fans is a class of spaces that have been studied extensively and are involved in the solutions of many problems that were seemingly unrelated. For one example, see .
Let be a non-trivial convergent sequence along with its limit point. For convenience, let , considered as a subspace of the Euclidean real line. Let be a cardinal number. The set is usually taken as the set of all the ordinals that precede . The set is the first infinite ordinal, or equivalently the set of all non-negative integers. Let be the set of all functions from into .
There are several ways to describe a sequential fan. One way is to describe it as a quotient space. The sequential fan is the topological sum of many copies of the convergent sequence with all non-isolated points identified as one point that is called . To make the discussion easier to follow, we also use the following formulation of :
In this formulation, every point is is isolated and an open neighborhood of the point is of the form:
According to the definition of the open neighborhood , the sequence converges to the point for each . Thus the set is a homeomorphic copy of the convergent sequence . The set is sometimes called a spine. Thus the space is said to be the sequential fan with many spines.
The point is the only non-isolated point in the fan . The set is a local base at the point . The base is never countable except when is finite. Thus if is infinite, the fan can never be first countable. In particular, for the fan , the character at the point is the cardinal number . See page 13 in chapter a-3 of . This cardinal number is called the dominating number and is introduced below in the section “The bounding number”. This is one indication that the sequential fan is highly dependent on set theory. It is hard to pinpoint the character of at the point . For example, it is consistent with ZFC that . It is also consistent that .
Even though the sequential fan is not first countable, it has a relatively strong convergent property. If and where , then infinitely many points of are present in at least one of the spine (if this is not true, then ). This means that the sequential fan is always a Frechet space. Recall that the space is a Frechet space if for each and for each , there exists a sequence of points of converging to .
Some of the convergent properties weaker than being a first countable space are Frechet space, sequential space and countably tight space. Let's recall the definitions. A space is a sequential space if is a sequentially closed set in , then is a closed set in . The set is sequentially closed in if this condition is satisfied: if the sequence converges to , then . A space is countably tight (have countable tightness) if for each and for each , there exists a countable such that . See here for more information about these convergent properties. The following shows the relative strength of these properties. None of the implications can be reversed.
First countable space Frechet space Sequential space Countably tight space
The relatively strong convergent property of being a Frechet space is not preserved in products or squares of sequential fans. We now look at some examples.
Consider the product space where is the convergent sequence defined above. The first factor is Frechet and the second factor is a compact metric space. We show that is not sequential. To see this, consider the following subset of :
It follows that . Furthermore, no sequence of points of can converge to the point . To see this, let for each . Consider two cases. One is that some spine contains the first coordinate of for infinitely many . The second is the opposite of the first – each spine contains the first coordinate of for at most finitely many . Either case means that there is an open set containing that misses infinitely many . Thus the sequence cannot converge to .
Let be the set of all sequential limits of convergent sequences of points of . With , we know that but . Thus is a sequentially closed subset of that is not closed. This shows that is not a sequential space.
The space is an example of a space that is countably tight but not sequential. The example shows that the product of two Frechet spaces does not even have to be sequential even when one of the factors is a compact metric space. The next example shows that the product of two sequential fans does not even have to be countably tight.
Consider the product space . We show that it is not countably tight. To this end, consider the following subset of .
It follows that . We show that for any countable , the point . Fix a countable . We can assume that . Now define a function by a diagonal argument as follows.
Define such that . For each integer , define such that and . Let . The diagonal definition of ensures that is an open set containing such that . This shows that the space is not countably tight.
The space is not countably tight. In fact its tightness character is . This fact follows from Theorem 1.1 in .
The set-theoretic angle
Example 2 shows that is not countably tight even though each factor has the strong property of a Frechet space with the first factor being a countable space. The example shows that Frechetness behaves very badly with respect to the product operation. Is there an example of such that is countably tight? In particular, is countably tight?
First off, if Continuum Hypothesis (CH) holds, then Example 2 shows that is not countably tight since the cardinality of is under CH. So for to be countably tight, extra set theory assumptions beyond ZFC will have to be used (in fact the extra axioms will have to be compatible with the negation of CH). In fact, it is consistent with ZFC for to be countably tight. It is also consistent with ZFC for . We point out some facts from the literature to support these observations.
Consider where . For any regular cardinal , it is possible that is countably tight. It is also possible for the tightness character of to be (of course in a different model of set theory). Thus it is hard to pin down the tightness character of the product . It all depends on your set theory. In the next section, we point out some facts from the literature to support these observations.
Example 3 points out that the tightness character of is , i.e. (this is a fact on the basis of ZFC only). What is or for any ? The tightness character of for also depends on set theory. We also give a brief explanation by pointing out some basic information from the literature.
The bounding number
The tightness of the product is related to the cardinal number called the bounding number denoted by .
Recall that is the set of all functions from into . For , define by the condition: for all but finitely many . A set is said to be a bounded set if has an upper bound according to , i.e. there exists some such that for all . Then is an unbounded set if it is not bounded. To spell it out, is an unbounded set if for each , there exists some such that .
Furthermore, is a dominating set if for each , there exists some such that . Define the cardinal numbers and as follows:
The cardinal number is called the bounding number. The cardinal number is called the dominating number. Note that continuum , the cardinality of , is an upper bound of both and , i.e. and . How do and relate? We have since any dominating set is also an unbounded set.
A diagonal argument (similar to the one in Example 2) shows that no countable can be unbounded. Thus we have . If CH holds, then we have . On the other hand, it is also consistent that .
We now relate the bounding number to the tightness of . The following theorem is from Theorem 1.3 in .
Theorem 1 – Theorem 1.3 in 
The following conditions hold:
- For , the space is countably tight.
- The tightness character of is , i.e. .
Thus is countably tight for any uncountable . In particular if , then is countably tight. According to Theorem 5.1 in , this is possible.
Theorem 2 – Theorem 5.1 in 
Let and be regular cardinal numbers such that . It is consistent with ZFC that and .
Theorem 2 indicates that it is consistent with ZFC that the bounding number can be made to equal any regular cardinal number. In the model of set theory in which , is countably tight. Likewise, in the model of set theory in which , is countably tight.
On the other hand, if the bounding number is made to equal an uncountable regular cardinal , then . In particular, if .
The above discussion shows that the tightness of is set-theoretic sensitive. Theorem 2 indicates that it is hard to pin down the location of the bounding number . Choose your favorite uncountable regular cardinal, there is always a model of set theory in which is your favorite uncountable cardinal. Then Theorem 1 ties the bounding number to the tightness of . Thus the exact value of the tightness character of depends on your set theory. If your favorite uncountable regular cardinal is , then in one model of set theory consistent with ZFC, (when ). In another model of set theory, (when ).
One comment about the character of the fan at the point . As indicated earlier, the character at is the dominating number . Theorem 2 tells us that it is consistent that can be any uncountable regular cardinal. So for the fan , it is quite difficult to pinpoint the status of a basic topological property such as character of a space. This is another indication that the sequential fan is highly dependent on additional axioms beyond ZFC.
The collectionwise Hausdorff property
Now we briefly discuss the tightness of for any . The following is Theorem 1.1 in .
Theorem 3 – Theorem 1.1 in 
Let be any infinite regular cardinal. The following conditions are equivalent.
- There exists a first countable -collectionwise Hausdorff space which fails to be a -collectionwise Hausdorff space.
The existence of the space in the first condition, on the surface, does not seem to relate to the tightness character of the square of a sequential fan. Yet the two conditions were proved to be equivalent . The existence of the space in the first condition is highly set-theory sensitive. Thus so is the tightness of the square of a sequential fan. It is consistent that a space in the first condition exists for . Thus in that model of set theory . It is also consistent that there does not exist a space in the first condition for . Thus in that model, . For more information, see .
Sequential fans and their products are highly set-theoretic in nature and are objects that had been studied extensively. This is only meant to be a short introduction. Any interested readers can refer to the small list of articles listed in the reference section and other articles in the literature.
Use Theorem 3 to show that by finding a space that is a first countable -collectionwise Hausdorff space which fails to be a -collectionwise Hausdorff space.
For any cardinal , a space is -collectionwise Hausdorff (respectively -collectionwise Hausdorff) if for any closed and discrete set with (repectively ), the points in can be separated by a pairwise disjoint family of open sets.
- Bella A., van Mill J., Tight points and countable fan-tightness, Topology Appl., 76, (1997), 1-27.
- Eda K., Gruenhage G., Koszmider P., Tamano K., Todorčeviće S., Sequential fans in topology, Topology Appl., 67, (1995), 189-220.
- Eda K., Kada M., Yuasa Y., Tamano K., The tightness about sequential fans and combinatorial properties, J. Math. Soc. Japan, 49 (1), (1997), 181-187.
- Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
- LaBerge T., Landver A., Tightness in products of fans and psuedo-fans, Topology Appl., 65, (1995), 237-255.
- Van Douwen, E. K., The Integers and Topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 111-167.