# 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 $\kappa$.

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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.

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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.

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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.

• The $\Sigma$-product of first countable spaces is a Frechet-Urysohn space.
• The $\Sigma$-product of separable metric spaces is monolithic.

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.

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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 $X$.

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Blog posts on Corson compact spaces

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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.

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