In this post, we introduce a class of hyperspaces called Pixley-Roy spaces. This is a well-known and well studied set of topological spaces. Our goal here is not to be comprehensive but rather to present some selected basic results to give a sense of what Pixley-Roy spaces are like.
A hyperspace refers to a space in which the points are subsets of a given “ground” space. There are more than one way to define a hyperspace. Pixley-Roy spaces were first described by Carl Pixley and Prabir Roy in 1969 (see ). In such a space, the points are the non-empty finite subsets of a given ground space. More precisely, let be a space (i.e. finite sets are closed). Let be the set of all non-empty finite subsets of . For each and for each open subset of with , we define:
The sets over all possible and form a base for a topology on . This topology is called the Pixley-Roy topology (or Pixley-Roy hyperspace topology). The set with this topology is called a Pixley-Roy space.
The hyperspace as defined above was first defined by Pixley and Roy on the real line (see ) and was later generalized by van Douwen (see ). These spaces are easy to define and is useful for constructing various kinds of counterexamples. Pixley-Roy played an important part in answering the normal Moore space conjecture. Pixley-Roy spaces have also been studied in their own right. Over the years, many authors have investigated when the Pixley-Roy spaces are metrizable, normal, collectionwise Hausdorff, CCC and homogeneous. For a small sample of such investigations, see the references listed at the end of the post. Our goal here is not to discuss the results in these references. Instead, we discuss some basic properties of Pixley-Roy to solidify the definition as well as to give a sense of what these spaces are like. Good survey articles of Pixley-Roy are  and .
In this section, we focus on properties that are always possessed by a Pixley-Roy space given that the ground space is at least . Let be a space. We discuss the following points:
- The topology defined above is a legitimate one, i.e., the sets indeed form a base for a topology on .
- is a Hausdorff space.
- is a zero-dimensional space.
- is a completely regular space.
- is a hereditarily metacompact space.
Let . Note that every finite set belongs to at least one set in , namely . So is a cover of . For , we have . So is indeed a base for a topology on .
To show is Hausdorff, let and be finite subsets of where . Then one of the two sets has a point that is not in the other one. Assume we have . Since is , we can find open sets such that , and . Then and are disjoint open sets containing and respectively.
To see that is a zero-dimensional space, we show that is a base consisting of closed and open sets. To see that is closed, let . Either or . In either case, we can choose open with such that .
The fact that is completely regular follows from the fact that it is zero-dimensional.
To show that is metacompact, let be an open cover of . For each , choose such that and let . Then is a point-finite open refinement of . For each , can only possibly belong to for the finitely many .
A similar argument show that is hereditarily metacompact. Let . Let be an open cover of . For each , choose such that and let . Then is a point-finite open refinement of . For each , can only possibly belong to for the finitely many such that .
More Basic Results
We now discuss various basic topological properties of . We first note that is a discrete space if and only if the ground space is discrete. Though we do not need to make this explicit, it makes sense to focus on non-discrete spaces when we look at topological properties of . We discuss the following points:
- If is uncountable, then is not separable.
- If is uncountable, then every uncountable subspace of is not separable.
- If is Lindelof, then is countable.
- If is Baire space, then is discrete.
- If has the CCC, then has the CCC.
- If has the CCC, then has no uncountable discrete subspaces,i.e., has countable spread, which of course implies CCC.
- If has the CCC, then is hereditarily Lindelof.
- If has the CCC, then is hereditarily separable.
- If has a countable network, then has the CCC.
- The Pixley-Roy space of the Sorgenfrey line does not have the CCC.
- If is a first countable space, then is a Moore space.
Bullet points 6 to 9 refer to properties that are never possessed by Pixley-Roy spaces except in trivial cases. Bullet points 6 to 8 indicate that can never be separable and Lindelof as long as the ground space is uncountable. Note that is discrete if and only if is discrete. Bullet point 9 indicates that any non-discrete can never be a Baire space. Bullet points 10 to 13 give some necessary conditions for to be CCC. Bullet 14 gives a sufficient condition for to have the CCC. Bullet 15 indicates that the hereditary separability and the hereditary Lindelof property are not sufficient conditions for the CCC of Pixley-Roy space (though they are necessary conditions). Bullet 16 indicates that the first countability of the ground space is a strong condition, making a Moore space.
To see bullet point 6, let be an uncountable space. Let be any countable subset of . Choose a point that is not in any . Then none of the sets belongs to the basic open set . Thus can never be separable if is uncountable.
To see bullet point 7, let be uncountable. Let . Let be any countable subset of . We can choose a point that is not in any . Choose some such that . Then none of the sets belongs to the open set . So not only is not separable, no uncountable subset of is separable if is uncountable.
To see bullet point 8, note that has no countable open cover consisting of basic open sets, assuming that is uncountable. Consider the open collection . Choose that is not in any of the sets . Then cannot belong to for any . Thus can never be Lindelof if is uncountable.
For an elementary discussion on Baire spaces, see this previous post.
To see bullet point 9, let be a non-discrete space. To show is not Baire, we produce an open subset that is of first category (i.e. the union of countably many closed nowhere dense sets). Let a limit point (i.e. an non-isolated point). We claim that the basic open set is a desired open set. Note that where
We show that each is closed and nowhere dense in the open subspace . To see that it is closed, let with . We have . Then is open and every point of has more than points of the space . To see that is nowhere dense in , let be open with . It is clear that where is open in the ground space . Since the point is not an isolated point in the space , contains infinitely many points of . So choose an finite set with at least points such that . For the the open set , we have and contains no point of . With the open set being a union of countably many closed and nowhere dense sets in , the open set is not of second category. We complete the proof that is not a Baire space.
To see bullet point 10, let be an uncountable and pairwise disjoint collection of open subsets of . For each , choose a point . Then is an uncountable and pairwise disjoint collection of open subsets of . Thus if is CCC then must have the CCC.
To see bullet point 11, let be uncountable such that as a space is discrete. This means that for each , there exists an open such that and contains no point of other than . Then is an uncountable and pairwise disjoint collection of open subsets of . Thus if has the CCC, then the ground space has no uncountable discrete subspace (such a space is said to have countable spread).
To see bullet point 12, let be uncountable such that is not Lindelof. Then there exists an open cover of such that no countable subcollection of can cover . We can assume that sets in are open subsets of . Also by considering a subcollection of if necessary, we can assume that cardinality of is or . Now by doing a transfinite induction we can choose the following sequence of points and the following sequence of open sets:
such that if , and for each . At each step , all the previously chosen open sets cannot cover . So we can always choose another point of and then choose an open set in that contains .
Then is a pairwise disjoint collection of open subsets of . Thus if has the CCC, then must be hereditarily Lindelof.
To see bullet point 13, let . Consider open sets where ranges over all finite subsets of and ranges over all open subsets of with . Let be a collection of such such that is pairwise disjoint and is maximal (i.e. by adding one more open set, the collection will no longer be pairwise disjoint). We can apply a Zorn lemma argument to obtain such a maximal collection. Let be the following subset of .
We claim that the set is dense in . Suppose that there is some open set such that and . Let . Then for all . So adding to , we still get a pairwise disjoint collection of open sets, contradicting that is maximal. So is dense in .
If has the CCC, then is countable and is a countable dense subset of . Thus if has the CCC, the ground space is hereditarily separable.
A collection of subsets of a space is said to be a network for the space if any non-empty open subset of is the union of elements of , equivalently, for each and for each open with , there is some with . Note that a network works like a base but the elements of a network do not have to be open. The concept of network and spaces with countable network are discussed in these previous posts Network Weight of Topological Spaces – I and Network Weight of Topological Spaces – II.
To see bullet point 14, let be a network for the ground space such that is also countable. Assume that is closed under finite unions (for example, adding all the finite unions if necessary). Let be a collection of basic open sets in . Then for each , find such that . Since is countable, there is some such that is uncountable. It follows that for any finite , .
Thus if the ground space has a countable network, then has the CCC.
The implications in bullet points 12 and 13 cannot be reversed. Hereditarily Lindelof property and hereditarily separability are not sufficient conditions for to have the CCC. See  for a study of the CCC property of the Pixley-Roy spaces.
To see bullet point 15, let be the Sorgenfrey line, i.e. the real line with the topology generated by the half closed intervals of the form . For each , let . Then is a collection of pairwise disjoint open sets in .
A Moore space is a space with a development. For the definition, see this previous post.
To see bullet point 16, for each , let be a decreasing local base at . We define a development for the space .
For each finite and for each , let . Clearly, the sets form a decreasing local base at the finite set . For each , let be the following collection:
We claim that is a development for . To this end, let be open in with . If we make large enough, we have .
For each non-empty proper , choose an integer such that and . Let be defined by:
We have for all non-empty proper . Thus for all non-empty proper . But in , the only sets that contain are and for all non-empty proper . So is the only set in that contains , and clearly .
We have shown that for each open in with , there exists an such that any open set in that contains must be a subset of . This shows that the defined above form a development for .
In the original construction of Pixley and Roy, the example was . Based on the above discussion, is a non-separable CCC Moore space. Because the density (greater than for not separable) and the cellularity ( for CCC) do not agree, is not metrizable. In fact, it does not even have a dense metrizable subspace. Note that countable subspaces of are metrizable but are not dense. Any uncountable dense subspace of is not separable but has the CCC. Not only is not metrizable, it is not normal. The problem of finding for which is normal requires extra set-theoretic axioms beyond ZFC (see ). In fact, Pixley-Roy spaces played a large role in the normal Moore space conjecture. Assuming some extra set theory beyond ZFC, there is a subset such that is a CCC metacompact normal Moore space that is not metrizable (see Example I in ).
On the other hand, Pixley-Roy space of the Sorgenfrey line and the Pixley-Roy space of (the first uncountable ordinal with the order topology) are metrizable (see ).
The Sorgenfrey line and the first uncountable ordinal are classic examples of topological spaces that demonstrate that topological spaces in general are not as well behaved like metrizable spaces. Yet their Pixley-Roy spaces are nice. The real line and other separable metric spaces are nice spaces that behave well. Yet their Pixley-Roy spaces are very much unlike the ground spaces. This inverse relation between the ground space and the Pixley-Roy space was noted by van Douwen (see  and ) and is one reason that Pixley-Roy hyperspaces are a good source of counterexamples.
- Bennett, H. R., Fleissner, W. G., Lutzer, D. J., Metrizability of certain Pixley-Roy spaces, Fund. Math. 110, 51-61, 1980.
- Daniels, P, Pixley-Roy Spaces Over Subsets of the Reals, Topology Appl. 29, 93-106, 1988.
- Lutzer, D. J., Pixley-Roy topology, Topology Proc. 3, 139-158, 1978.
- Hajnal, A., Juahasz, I., When is a Pixley-Roy Hyperspace CCC?, Topology Appl. 13, 33-41, 1982.
- Pixley, C., Roy, P., Uncompletable Moore spaces, Proc. Auburn Univ. Conf. Auburn, AL, 1969.
- Przymusinski, T., Normality and paracompactness of Pixley-Roy hyperspaces, Fund. Math. 113, 291-297, 1981.
- van Douwen, E. K., The Pixley-Roy topology on spaces of subsets, Set-theoretic Topology, Academic Press, New York, 111-134, 1977.
- Tall, F. D., Normality versus Collectionwise Normality, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 685-732, 1984.
- Tanaka, H, Normality and hereditary countable paracompactness of Pixley-Roy hyperspaces, Fund. Math. 126, 201-208, 1986.