This post is a basic discussion on countably paracompact space. A space is a paracompact space if every open cover has a locally finite open refinement. The definition can be tweaked by saying that only open covers of size not more than a certain cardinal number can have a locally finite open refinement (any space with this property is called a -paracompact space). The focus here is that the open covers of interest are countable in size. Specifically, a space is a countably paracompact space if every countable open cover has a locally finite open refinement. Even though the property appears to be weaker than paracompact spaces, the notion of countably paracompactness is important in general topology. This post discusses basic properties of such spaces. All spaces under consideration are Hausdorff.
A related notion is that of metacompactness. A space is a metacompact space if every open cover has a point-finite open refinement. For a given open cover, any locally finite refinement is a point-finite refinement. Thus paracompactness implies metacompactness. The countable version of metacompactness is also interesting. A space is countably metacompact if every countable open cover has a point-finite open refinement. In fact, for any normal space, the space is countably paracompact if and only of it is countably metacompact (see Corollary 2 below).
Normal Countably Paracompact Spaces
A good place to begin is to look at countably paracompactness along with normality. In 1951, C. H. Dowker characterized countably paracompactness in the class of normal spaces.
Theorem 1 (Dowker’s Theorem)
Let be a normal space. The following conditions are equivalent.
- The space is countably paracompact.
- Every countable open cover of has a point-finite open refinement.
- If is an open cover of , there exists an open refinement such that for each .
- The product space is normal for any compact metric space .
- The product space is normal where is the closed unit interval with the usual Euclidean topology.
- For each sequence of closed subsets of such that and , there exist open sets such that for each such that .
Dowker’s Theorem is proved in this previous post. Condition 2 in the above formulation of the Dowker’s theorem is not in the Dowker’s theorem in the previous post. In the proof for in the previous post is essentially for Theorem 1 above. As a result, we have the following.
Let be a normal space. Then is countably paracompact if and only of is countably metacompact.
Theorem 1 indicates that normal countably paracompact spaces are important for the discussion of normality in product spaces. As a result of this theorem, we know that normal countably paracompact spaces are productively normal with compact metric spaces. The Cartesian product of normal spaces with compact spaces can be non-normal (an example is found here). When the normal factor is countably paracompact and the compact factor is upgraded to a metric space, the product is always normal. The connection with normality in products is further demonstrated by the following corollary of Theorem 1.
Let be a normal space. Let be a non-discrete metric space. If is normal, then is countably paracompact.
Since is non-discrete, there is a non-trivial convergent sequence (i.e. the sequence represents infinitely many points). Then the sequence along with the limit point is a compact metric subspace of . Let’s call this subspace . Then is a closed subspace of the normal . As a result, is normal. By Theorem 1, is countably paracompact.
C. H. Dowker in 1951 raised the question: is every normal space countably compact? Put it in another way, is the product of a normal space and the unit interval always a normal space? As a result of Theorem 1, any normal space that is not countably paracompact is called a Dowker space. The search for a Dowker space took about 20 years. In 1955, M. E. Rudin showed that a Dowker space can be constructed from assuming a Souslin line. In the mid 1960s, the existence of a Souslin line was shown to be independent of the usual axioms of set theorey (ZFC). Thus the existence of a Dowker space was known to be consistent with ZFC. In 1971, Rudin constructed a Dowker space in ZFC. Rudin’s Dowker space has large cardinality and is pathological in many ways. Zoltan Balogh constructed a small Dowker space (cardinality continuum) in 1996. Various Dowker space with nicer properties have also been constructed using extra set theory axioms. The first ZFC Dowker space constructed by Rudin is found in . An in-depth discussion of Dowker spaces is found in . Other references on Dowker spaces is found in .
Since Dowker spaces are rare and are difficult to come by, we can employ a “probabilistic” argument. For example, any “concrete” normal space (i.e. normality can be shown without using extra set theory axioms) is likely to be countably paracompact. Thus any space that is normal and not paracompact is likely countably paracompact (if the fact of being normal and not paracompact is established in ZFC). Indeed, any well known ZFC example of normal and not paracompact must be countably paracompact. In the long search for Dowker spaces, researchers must have checked all the well known examples! This probability thinking is not meant to be a proof that a given normal space is countably paracompact. It is just a way to suggest a possible answer. In fact, a good exercise is to pick a normal and non-paracompact space and show that it is countably paracompact.
The following lists out a few classes of spaces that are always countably paracompact.
- Metric spaces are countably paracompact.
- Paracompact spaces are countably paracompact.
- Compact spaces are countably paracompact.
- Countably compact spaces are countably paracompact.
- Perfectly normal spaces are countably paracompact.
- Normal Moore spaces are countably paracompact.
- Linearly ordered spaces are countably paracompact.
- Shrinking spaces are countably paracompact.
The first four bullet points are clear. Metric spaces are paracompact. It is clear from definition that paracompact spaces, compact and countably compact spaces are countably paracompact. One way to show perfect normal spaces are countably paracompact is to show that they satisfy condition 6 in Theorem 1 (shown here). Any Moore space is perfect (closed sets are ). Thus normal Moore space are perfectly normal and hence countably paracompact. The proof of the countably paracompactness of linearly ordered spaces can be found in . See Theorem 5 and Corollary 6 below for the proof of the last bullet point.
As suggested by the probability thinking in the last section, we now look at examples of countably paracompact spaces among spaces that are “normal and not paracompact”. The first uncountable ordinal is normal and not paracompact. But it is countably compact and is thus countably paracompact.
Any -product of uncountably many metric spaces is normal and countably paracompact.
For each , let be a metric space that has at least two points. Assume that each has a point that is labeled 0. Consider the following subspace of the product space .
The space is said to be the -product of the spaces . It is well known that the -product of metric spaces is normal, in fact collectionwise normal (this previous post has a proof that -product of separable metric spaces is collectionwise normal). On the other hand, any -product always contains as a closed subset as long as there are uncountably many factors and each factor has at least two points (see the lemma in this previous post). Thus any such -product, including the one being discussed, cannot be paracompact.
Next we show that is normal. The space can be reformulated as a -product of metric spaces and is thus normal. Note that where , for any with , and for any with , . Thus is normal since it is the -product of metric spaces. By Theorem 1, the space is countably paracompact.
Let be any uncountable cardinal number. Let be the discrete space of cardinality . Let be the one-point Lindelofication of . This means that where is a point not in . In the topology for , points in are isolated as before and open neighborhoods at are of the form where is any countable subset of . Now consider , the space of real-valued continuous functions defined on endowed with the pointwise convergence topology. The space is normal and not Lindelof, hence not paracompact (discussed here). The space is also homeomorphic to a -product of many copies of the real lines. By the same discussion in Example 1, is countably paracompact. For the purpose at hand, Example 2 is similar to Example 1.
Consider R. H. Bing’s example G, which is a classic example of a normal and not collectionwise normal space. It is also countably paracompact. This previous post shows that Bing’s Example G is countably metacompact. By Corollary 2, it is countably paracompact.
Based on the “probabilistic” reasoning discussed at the end of the last section (based on the idea that Dowker spaces are rare), “normal countably paracompact and not paracompact” should be in plentiful supply. The above three examples are a small demonstration of this phenomenon.
Existence of Dowker spaces shows that normality by itself does not imply countably paracompactness. On the other hand, paracompact implies countably paracompact. Is there some intermediate property that always implies countably paracompactness? We point that even though collectionwise normality is intermediate between paracompactness and normality, it is not sufficiently strong to imply countably paracompactness. In fact, the Dowker space constructed by Rudin in 1971 is collectionwise normal.
More on Countably Paracompactness
Without assuming normality, the following is a characterization of countably paracompact spaces.
Let be a topological space. Then the space is countably paracompact if and only of the following condition holds.
- For any decreasing sequence of closed subsets of such that , there exists a decreasing sequence of open subsets of such that for each and .
Proof of Theorem 4
Suppose that is countably paracompact. Suppose that is a decreasing sequence of closed subsets of as in the condition in the theorem. Then is an open cover of . Let be a locally finite open refinement of . For each , define the following:
It is clear that for each . The open sets are decreasing, i.e. since the closed sets are decreasing. To show that , let . The goal is to find such that . Once is found, we will obtain an open set such that and contains no points of .
Since is locally finite, there exists an open set such that and meets only finitely many sets in . Suppose that these finitely many open sets in are . Observe that for each , there is some such that (i.e. ). This follows from the fact that is a refinement . Let be the maximum of all where . Then for all . It follows that the open set contains no points of . Thus .
For the other direction, suppose that the space satisfies the condition given in the theorem. Let be an open cover of . For each , define as follows:
Then the closed sets form a decreasing sequence of closed sets with empty intersection. Let be decreasing open sets such that and for each . Let for each . Then . Define . For each , define . Clearly each is open and . It is straightforward to verify that is a cover of .
We claim that is locally finite in . Let . Choose the least such that . Choose an open set such that and . Then and . This means that for all . Thus the open cover is a locally finite refinement of .
We present another characterization of countably paracompact spaces that involves the notion of shrinkable open covers. An open cover of a space is said to be shrinkable if there exists an open cover of the space such that for each , . If is shrinkable by , then we also say that is a shrinking of . Note that Theorem 1 involves a shrinking. Condition 3 in Theorem 1 (Dowker’s Theorem) can rephrased as: every countable open cover of has a shrinking. This for any normal countably paracompact space, every countable open cover has a shrinking (or is shrinkable).
A space is a shrinking space if every open cover of is shrinkable. Every shrinking space is a normal space. This follows from this lemma: A space is normal if and only if every point-finite open cover of is shrinkable (see here for a proof). With this lemma, it follows that every shrinking space is normal. The converse is not true. To see this we first show that any shrinking space is countably paracompact. Since any Dowker space is a normal space that is not countably paracompact, any Dowker space is an example of a normal space that is not a shrinking space. To show that any shrinking space is countably paracompact, we first prove the following characterization of countably paracompactness.
Let be a space. Then is countably paracompact if and only of every countable increasing open cover of is shrinkable.
Proof of Theorem 5
Suppose that is countably paracompact. Let be an increasing open cover of . Then there exists a locally open refinement of . For each , define . Then is also a locally finite refinement of . For each , define
Let . It follows that if . Then is an increasing open cover of . Observe that for each , for all . Then we have the following:
We have just established that is a shrinking of , or that is shrinkable.
For the other direction, to show that is countably paracompact, we show that the condition in Theorem 4 is satisfied. Let be a decreasing sequence of closed subsets of with empty intersection. Then be an open cover of where for each . By assumption, is shrinkable. Let be a shrinking. We can assume that is an increasing sequence of open sets.
For each , let . We claim that is a decreasing sequence of open sets that expand the closed sets and that . The expansion part follows from the following:
The part about decreasing follows from:
We show that . To this end, let . Then for some . We claim that . Suppose . Since is an open set containing , must contain a point of , say . Since , . This in turns means that , a contradiction. Thus we have as claimed. We have established that every point of is not in for some . Thus the intersection of all the must be empty. We have established the condition in Theorem 4 is satisfied. Thus is countably paracompact.
If is a shrinking space, then is countably paracompact.
- Ball, B. J., Countable Paracompactness in Linearly Ordered Spaces, Proc. Amer. Math. Soc., 5, 190-192, 1954. (link)
- Rudin, M. E., A Normal Space for which is not Normal, Fund. Math., 73, 179-486, 1971. (link)
- Rudin, M. E., Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 761-780.
- Wikipedia Entry on Dowker Spaces (link)