Basic topological properties of Corson compact spaces

A compact space is said to be a Corson compact space if it can be embedded in a \Sigma-product of real lines. Corson compact spaces play an important role in functional analysis. Corson compact spaces are also very interesting from a topological point of view. In this post, we discuss several basic topological properties of Corson compact spaces, some of which have been discussed and proved in some previous posts.

Links to other posts on Corson compact spaces are given throughout this post (as relevant results are discussed). A listing of these previous posts is also given at the end of this post.

For any infinite cardinal number \kappa, the \Sigma-product of \kappa many copies of \mathbb{R} is the following subspace of the product space \mathbb{R}^\kappa:

    \Sigma(\kappa)=\left\{x \in \mathbb{R}^\kappa: x_\alpha \ne 0 \text{ for at most countably many } \alpha < \kappa \right\}

A compact space is said to be a Corson compact space if it can be embedded in \Sigma(\kappa) for some infinite cardinal \kappa.

____________________________________________________________________

Some examples of Corson compact spaces

When \kappa=\omega, \Sigma(\omega) is simply \mathbb{R}^\omega, the product of countably many copies of the real lines. Any compact metrizable space can be embedded in \mathbb{R}^\omega; see Theorem 4.2.10 in [3]. Thus any compact metrizable space is Corson compact.

One easily described non-metrizable Corson compact space is the one-point compactification of an uncountable discrete space. Let \tau be an uncountable cardinal number. Let D_\tau be the discrete space of cardinality \tau. Let p be a point not in D_\tau. Let A_\tau=D_\tau \cup \left\{p \right\}. Consider a topology on A_\tau such that D_\tau is discrete as before and any open neighborhood of p has the form \left\{p \right\} \cup C where C \subset D_\tau and D_\tau-C is finite. The space A_\tau is better known as the one-point compactification of a discrete space of cardinality \tau. To see that A_\tau is embedded in \Sigma(\tau), for each \alpha<\tau, define f_\alpha:\tau \rightarrow \left\{0,1 \right\} such that f_\alpha(\alpha)=1 and f_\alpha(\beta)=0 for all \beta \ne \alpha. Furthermore, let g:\tau \rightarrow \left\{0,1 \right\} be defined by g(\beta)=0 for all \beta<\tau. It is easy to clear that K=\left\{f_\alpha: \alpha<\tau \right\} \cup \left\{g \right\} is a subspace of \Sigma(\tau). Note that \left\{f_\alpha: \alpha<\tau \right\} is a discrete space of cardinality \tau and that any open neighborhood of g contains all but finitely many f_\alpha. Thus K is homeomorphic to A_\tau.

Another non-metrizable Corson compact space is defined and discussed in this previous post.

____________________________________________________________________

Some basic operations

Corson compact spaces behave nicely with respect to some natural topological operations such as the following:

  1. Corson compactness is hereditary with respect to closed subspaces.
  2. Corson compactness is preserved by taking continuous images.
  3. Corson compactness is preserved by taking countable products.

It is clear that closed subspaces of every Corson compact space are also Corson compact. The fact that any continuous image of a Corson compact space is Corson compact (bullet point #2) is established in Theorem 6.2 of [6].

We prove bullet point #3, that the Tychonoff product of countably many Corson compact spaces is Corson compact. Let K_1,K_2,K_3,\cdots be compact spaces such that each K_j is a subspace of \Sigma(\tau_j) for some infinite cardinal \tau_j. We show that the product \prod_{j=1}^\infty K_j is Corson compact. Let \tau be the cardinality of \bigcup_{j=1}^\infty \tau_j. Note that \prod_{j=1}^\infty K_j is a compact subspace of \prod_{j=1}^\infty \Sigma(\tau_j). In turn, \prod_{j=1}^\infty \Sigma(\tau_j) is identical to \Sigma(\tau). Thus \prod_{j=1}^\infty K_j is Corson compact.

____________________________________________________________________

Basic properties inherited from \Sigma-products

Some properties of Corson compact spaces are inherited from the \Sigma-products of real line. For example, we have the following two facts about \Sigma-products.

Both of the above two properties are hereditary. Thus we have the following:

  1. Every Corson compact space is a Frechet-Urysohn space.
  2. Every Corson compact space is monolithic.

A space X is monolithic if for each subspace Y of X, the density of Y coincides with the network weight of Y, i.e., d(Y)=nw(Y). A space X is strongly monolithic if for each subspace Y of X, the density of Y coincides with the weight of Y, i.e., d(Y)=w(Y). Monolithic spaces are discussed in this previous post. For compact spaces, the notion of being monolithic and the notion of being strongly monolithic coincide. One obvious consequence of bullet point #5 is that being separable is an indicator of metrizability among Corson compact spaces. The following bullet point captures this observation.

  1. Let X be a Corson compact space. Then X is metrizable if and only if X is separable. See Proposition 1 in this previous post.

A space Y is said to be Frechet space (also called Frechet-Urysohn space) if for each y \in Y and for each M \subset Y, if y \in \overline{M}, then there exists a sequence \left\{y_n \in M: n=1,2,3,\cdots \right\} such that the sequence converges to y. Thus any compact space that is not a Frechet-Urysohn space is not Corson compact. A handy example is the compact space \omega_1+1 with the order topology. Note that \omega_1+1 is monolithic. Thus monolithic compact spaces need not be Corson compact.

An extreme example of a compact non-Frechet-Urysohn space is one that has no non-trivial convergent sequence. For example, take \beta \omega, the Stone-Cech compactification of a countable discrete space, which has no non-trivial convergent sequences at any point. Thus \beta \omega is not Corson compact.

Every Corson compact space has a G_\delta point. It then follows that every Corson compact space has a dense set of G_\delta points (see this previous post). In a compact space, there is a countable local base at every G_\delta point. Thus we have the following bullet point.

  1. Every Corson compact space has a dense first countable subspace.

However, it is not true that every Corson compact space has a dense metrizable subspace. See Theorem 9.14 in [7] for an example of a first countable Corson compact space with no dense metrizable subspace.

____________________________________________________________________

Remarks

The results indicated here and proved in the previous posts represent a small sample of results on Corson compact spaces (and just focusing on the topological aspects). Many results on Corson compact spaces and Eberlein compact spaces are very deep results. The chapter c-16 in [5] is a good introduction. Some of the results proven in this and other posts in this blog are mentioned in [5] without proof. Interesting characterizations of Corson compact spaces are presented in [4].

In closing, we mention one more property. The authors in [1] showed that for any Corson compact space X, the function space C_p(X) with the pointwise convergence topology is a Lindelof space. Thus we have the following bullet point.

  1. The function space C_p(X) is Lindelof for every Corson compact space X.

____________________________________________________________________

Blog posts on Corson compact spaces

____________________________________________________________________

Reference

  1. Alster, K., Pol, R., On function spaces of compact subspaces of \Sigma-products of the real line, Fund. Math., 107, 35-46, 1980.
  2. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
  3. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  4. Gruenhage, G., Covering properties on X^2 \backslash \Delta, W-sets, and compact subsets of \Sigma-products, Topology Appl., 17, 287-304, 1984.
  5. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
  6. Michael, E., Rudin, M. E., A note on Eberlein compacts, Pacific J. Math., 128, 149-156 1987.
  7. Todorcevic, S., Trees and Linearly Ordered Sets, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 235-293, 1984.

____________________________________________________________________
\copyright \ 2014 \text{ by Dan Ma}

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s