It is well known that when and are paracompact spaces, the product space is not necessarily normal. Classic examples include the product of the Sorgenfrey line with itself (discussed here) and the product of the Michael line and the space of irrational numbers (discussed here). However, if one of the paracompact factors is “compact”, the product can be normal or even paracompact. This post discusses several classic results along this line. All spaces are Hausdorff and regular.
Suppose that and are paracompact spaces. We have the following results:
- If is a compact space, then is paracompact.
- If is a -compact space, then is paracompact.
- If is a locally compact space, then is paracompact.
- If is a -locally compact space, then is paracompact.
The proof of the first result makes uses the tube lemma. The second result is a corollary of the first. The proofs of both results are given here. The third result is a corollary of the fourth result. We give a proof of the fourth result.
Proof of the Fourth Result
The fourth result indicated above is restated as Theorem 2 below. It is a theorem of K. Morita . This is one classic result on product of paracompact spaces. After proving the theorem, comments are made about interesting facts and properties that follow from this result. Theorem 2 is also Theorem 3.22 in chapter 18 in the Handbook of Set-Theoretic Topology .
A space is a locally compact space if for each , there is an open subset of such that and is compact. When we say is a -locally compact space, we mean that where each is a locally compact space. In proving the result discussed here, we also assume that each is a closed subspace of . The following lemma will be helpful.
Let be a paracompact space. Suppose that is -locally compact. Then there exists a cover of such that each is a locally finite family consisting of compact sets.
Proof of Lemma 1
Let such that each is closed and is locally compact. Fix an integer . For each , let be an open subset of such that and is compact (the closure is taken in ). Consider the open cover of . Since is a closed subspace of , is also paracompact. Let be a locally finite open cover of such that for each (again the closure is taken in ). Each is compact since . Let .
We claim that is a locally finite family with respect to the space . For each , is an open set containing that intersects no set in . For each , there is an open set that meets only finitely many sets in . Extend to an open subset of . That is, is an open subset of such that . It is clear that can only meets finitely many sets in .
Then is the desired -locally finite cover of .
Let be any paracompact space and let be any -locally compact paracompact space. Then is paracompact.
Proof of Theorem 2
By Lemma 1, let be a -locally finite cover of such that each consists of compact sets. To show that is paracompact, let be an open cover of . For each and for each , the set is obviously compact.
Fix and fix . For each , the point for some . Choose open and open such that . Letting vary, the open sets cover the compact set . Choose finitely many open sets that also cover . Let be the intersection of these finitely many . Let be the set of these finitely many .
To summarize what we have obtained in the previous paragraph, for each and for each , there exists an open subset containing , and there exists a finite set of open subsets of such that
- for each , for some .
For each , the set of all is an open cover of . Since is paracompact, for each , there exists a locally finite open cover such that for all . Consider the following families of open sets.
We claim that is a -locally finite open refinement of . First, show that is an open cover of . Let . Then for some , for some . Furthermore, for some . The information about and are detailed above. For example, . Thus there exists some such that . We now have .
Next we show that is a refinement of . Fix . Immediately we see that . Since , for some . Then .
The remaining point to make is that each is a locally finite family of open subsets of . Let . Since is locally finite in , there exists some open such that and meets only finitely many sets in , say . Recall that is the set of all and is locally finite. Thus there exists an open such that and meets only finitely many sets in each where . Thus the open set meets only finitely many sets for finitely many and finitely many . These finitely many and lead to finitely many . Thus it follows that meets only finitely many sets in . Thus is locally finite.
What has been established is that every open cover of has a -locally finite open refinement. This fact is equivalent to paracompactness (according to Theorem 1 in this previous post). This concludes the proof of the theorem.
Productively Paracompact Spaces
Consider this property for a space .
(*) The space satisfies the property that is a paracompact space for every paracompact space .
Such a space can be called a productively paracompact space (for some reason, this term is not used in the literature).
According to the four results stated at the beginning, any space in any one of the following four classes
- Compact spaces.
- -compact spaces.
- Locally compact paracompact spaces.
- -locally compact paracompact spaces.
satisfies this property. Both the Michael line and the space of the irrational numbers are examples of paracompact spaces that do not have this productively paracompact property. According to comments made on page 799 , the theorem of Morita (Theorem 2 here) triggered extensive research to investigate this class of spaces. The class of spaces is broader than the four classes listed here. For example, the productively paracompact spaces also include the closed images of locally compact paracompact spaces. The handbook  has more references.
Consider this property.
(**) The space satisfies the property that is a normal space for every metric space .
These spaces can be called productively normal spaces with respect to metric spaces. They go by another name. Morita defined the notion of P-spaces and proved that a space is a normal P-space if and only if the product of with any metric space is normal.
Since the class of metric spaces contain the paracompact spaces, any space has property (*) would have property (**), i.e. a normal P-space.Thus any locally compact paracompact space is a normal P-space. Any -locally compact paracompact space is a normal P-space. If a paracompact space has any one of the four “compact” properties discussed here, it is a normal P-space.
Looking at Diagrams
Let’s compare these classes of spaces: productively paracompact spaces (the spaces satisfying property (*)), normal P-spaces and paracompact spaces. We have the following diagram.
Clearly productively paracompact implies paracompact. As discussed in the previous section, productively paracompact implies normal P. If a space is such that the product of with every paracompact space is paracompact, then the product of with every metric space is paracompact and hence normal.
However, the arrows in Diagram 1 are not reversible. The Michael line mentioned at the beginning will shed some light on this point. Here’s the previous post on Michael line. Let be the Michael line. Let be the space of the irrational numbers. The space would be a paracompact space that is not productively paracompact since its product with is not normal, hence not paracompact.
On the other hand, the space of irrational numbers is a normal P-space since it is a metric space. But it is not productively paracompact since its product with the Michael line is not normal, hence not paracompact.
The two classes of spaces at the bottom of Diagram 1 do not relate. The Michael line is a paracompact space that is not a normal P-space since its product with is not normal. Normal P-space does not imply paracompact. Any space that is normal and countably compact is a normal P-space. For example, the space , the first uncountable ordinal, with the ordered topology is normal and countably compact and is not paracompact.
There are other normal P-spaces that are not paracompact. For example, Bing’s Example H is perfectly normal and not paracompact. As mentioned in the previous section, any perfectly normal space is a normal P-space.
The class of spaces whose product with every paracompact space is paracompact is stronger than both classes of paracompact spaces and normal P-spaces. It is a strong property and an interesting class of spaces. It is also an excellent topics for any student who wants to dig deeper into paracompact spaces.
Let’s add one more property to Diagram 1.
All properties in Diagram 2 except for paracompact are productive. Normal countably paracompact spaces are productive. According to Dowker’s theorem, the product of any normal countably paracompact space with any compact metric space is normal (see Theorem 1 in this previous post). The last two arrows in Diagram 2 are also not reversible.
- Morita K., On the Product of Paracompact Spaces, Proc. Japan Acad., Vol. 39, 559-563, 1963.
- Przymusinski T. C., Products of Normal Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 781-826, 1984.
2017 – Dan Ma