This post is an introduction to countable tight and countably generated spaces. A space being a countably tight space is a convergence property. The article  lists out 8 convergence properties. The common ones on that list include Frechet space, sequential space, k-space and countably tight space, all of which are weaker than the property of being a first countable space. In this post we discuss several ways to define countably tight spaces and to discuss its generalizations.
A space is countably tight (or has countable tightness) if for each and for each , there is a countable such that . According to this Wikipedia entry, a space being a countably generated space is the property that its topology is generated by countable sets and is equivalent to the property of being countably tight. The equivalence of the two definitions is not immediately clear. In this post, we examine these definitions more closely. Theorem 1 below has three statements that are equivalent. Any one of the three statements can be the definition of countably tight or countably generated.
Let be a space. The following statements are equivalent.
- For each , the set equality (a) holds.
For all countable , is closed in
then is closed.
For all countable ,
then is closed.
Statement 1 is the definition of a countably tight space. The set inclusion in (a) is always true. We only need to be concerned with , which is the definition of countable tightness given earlier.
Statement 2 is the definition of a countably generated space according to this Wikipedia entry. This definition is in the same vein as that of k-space (or compactly generated space). Note that a space is a k-space if Statement 2 holds when “countable” is replaced with “compact”.
Statement 3 is in the same vein as that of a sequential space. Recall that a space is a sequential space if is a sequentially closed set then is closed. The set is a sequentially closed set if the sequence converges to , then (in other words, for any sequence of points of that converges, the limit must be in ). If the sequential limit in the definition of sequential space is relaxed to be just topological limit (i.e. accumulation point), then the resulting definition is Statement 3. Thus Statement 3 says that for any countable subset of , any limit point (i.e. accumulation point) of must be in . Thus any sequential space is countably tight. In a sequential space, the closed sets are generated by taking sequential limit. In a space defined by Statement 3, the closed sets are generated by taking closures of countable sets.
All three statements are based on the countable cardinality and have obvious generalizations by going up in cardinality. For any set that satisfies condition (c) in Statement 3 is said to be an -closed set. Thus for any cardinal number , the set is a -closed set if for any with , . Condition (c) in Statement 3 can then be generalized to say that if is a -closed set, then is closed.
The proof of Theorem 1 is handled in the next section where we look at the generalizations of all three statements and prove their equivalence.
The definition in Statement 1 in Theorem 1 above can be generalized as a cardinal function called tightness. Let be a space. By we mean the least infinite cardinal number such that the following holds:
For all , and for each , there exists with such that .
When , the space is countably tight (or has countable tightness). In keeping with the set equality (a) above, the tightness can also be defined as the least infinite cardinal such that for any , the following set equality holds:
Let be an infinite cardinal number. To generalize Statement 2, we say that a space is -generated if the following holds:
For each , if the following condition holds:
For all with , the set is closed in
then is closed.
To generalize Statement 3, we say that a set is -closed if for any with , . A generalization of Statement 3 is that
For any , if is a -closed set, then is closed
Let be a space. Let be an infinite cardinal. The following statements are equivalent.
- The space is -generated.
- For each , if is a -closed set, then is closed.
Proof of Theorem 2
Suppose that (2) does not hold. Let be such that the set satisfies condition and is not closed. Let . By (1), the point belongs to the right hand side of the set equality . Choose with such that . Let . By condition , is closed in . This would mean that and hence , a contradiction. Thus if (1) holds, (2) must holds.
Suppose (3) does not hold. Let be a -closed set that is not a closed set in . Since (2) holds and is not closed, condition must not hold. Choose with such that is not closed in . Choose that is in the closure of but is not in . Since is -closed, , which implies that , a contradiction. Thus if (2) holds, (3) must hold.
Suppose (1) does not hold. Let be such that the set equality does not hold. Let be such that does not belong to the right hand side of . Let . Note that the set is -closed. By (3), is closed. Furthermore , leading to , a contradiction. So if (3) holds, (1) must hold.
Theorem 1 obviously follows from Theorem 2 by letting . There is another way to characterize the notion of tightness using the concept of free sequence. See the next post.
Several elementary convergence properties have been discussed in a series of blog posts (the first post and links to the other are found in the first one). We have the following implications and none is reversible.
First countable Frechet Sequential k-space
Where does countable tightness place in the above implications? We discuss above that
Sequential countably tight.
How do countably tight space and k-space compare? It turns out that none implies the other. We present some supporting examples.
The Arens’ space is a canonical example of a sequential space that is not a Frechet space. A subspace of the Arens’ space is countably tight and not sequential. The same subspace is also not a k-space. There are several ways to represent the Arens’ space, we present the version found here.
Let be the set of all positive integers. Define the following:
Let . Each point in is an isolated point. Open neighborhoods at are of the form:
The open neighborhoods at are obtained by removing finitely many from and by removing finitely many isolated points in the that remain. The open neighborhoods just defined form a base for a topology on the set , i.e. by taking unions of these open neighborhoods, we obtain all the open sets for this space. The space can also be viewed as a quotient space (discussed here).
The space is a sequential space that is not Frechet. The subspace is not sequential. Since is a countable space, the space is by default a countably tight space. The space is also not an k-space. These facts are left as exercises below.
Consider the product space . The space is compact since it is a product of compact spaces. Any compact space is a k-space. Thus is a k-space (or compactly generated space). On the other hand, is not countably tight. Thus the notion of k-space and the notion of countably tight space do not relate.
There is another way to characterize the notion of tightness using the concept of free sequence. See the next post.
The notion of tightness had been discussed in previous posts. One post shows that the function space is countably tight when is compact (see here). Another post characterizes normality of when is compact (see here)
This is to verify Example 1. Verify that
- The space is a sequential space that is not Frechet.
- is not sequential.
- The space is not an k-space.
Verify that any compact space is a k-space. Show that the space in Example 2 is not countably tight.
- Gerlits J., Nagy Z., Products of convergence properties, Commentationes Mathematicae Universitatis Carolinae, Vol 23, No 4 (1982), 747–756