Sigma-products of separable metric spaces are monolithic

Let \Sigma(\kappa) be the \Sigma-product of \kappa many copies of the real lines where \kappa is any infinite cardinal number. Any compact space that can be embedded in \Sigma(\kappa) for some \kappa is said to be a Corson compact space. Corson compact spaces play an important role in functional analysis. Corson compact spaces are also very interesting from a topological point of view. Some of the properties of Corson compact spaces are inherited (as subspaces) from the \Sigma-product \Sigma(\kappa). One such property is the property that the \Sigma-product \Sigma(\kappa) is monolithic, which implies that the closure of any countable subspace of \Sigma(\kappa) is metrizable.

Previous blog posts on \Sigma-products:

A previous blog post on monolithic spaces: A short note on monolithic spaces. A listing of other blog posts on Corson compact spaces is given at the end of this post.

____________________________________________________________________

Defining Sigma-product

Let \kappa be an infinite cardinal number. For each \alpha<\kappa, let X_\alpha be a topological space. Let b \in \prod_{\alpha<\kappa} X_\alpha. The \Sigma-product of the spaces X_\alpha about the base point b is defined as follows:

    \Sigma_{\alpha<\kappa} X_\alpha=\left\{x \in \prod_{\alpha<\kappa} X_\alpha: x_\alpha \ne b_\alpha \text{ for at most countably many } \alpha < \kappa \right\}

If each X_\alpha=\mathbb{R} and if the base point b is such that b_\alpha=0 for all \alpha<\kappa, then we use the notation \Sigma(\kappa) for \Sigma_{\alpha<\kappa} X_\alpha, i.e., \Sigma(\kappa) is defined as follows:

    \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 the \Sigma-product \Sigma(\kappa) for some infinite cardinal \kappa.

____________________________________________________________________

Monolithic Spaces

A space X is monolithic if for every subspace Y of X, the density of Y equals the network weight of Y, i.e., d(Y)=nw(Y). A space X is strongly monolithic if for every subspace Y of X, the density of Y equals the weight of Y, i.e., d(Y)=w(Y). See the previous post called A short note on monolithic spaces.

The proof of the fact that \Sigma-product of separable metrizable spaces is monolithic can be worked out quite easily from definitions. Interested readers are invited to walk through the proof. For the sake of completeness, we prove the following theorem.

Theorem 1
Suppose that for each \alpha<\kappa, X_\alpha is a separable metric space. Then the \Sigma-product \Sigma_{\alpha<\kappa} X_\alpha is strongly monolithic.

Proof of Theorem 1
Let b be the base point of the \Sigma-product X=\Sigma_{\alpha<\kappa} X_\alpha. For each x \in X, let S(x) be the support of the point x, i.e., the set of all \alpha<\kappa such that x_\alpha \ne b_\alpha. Let Y be a subspace of X. We show that d(Y)=w(Y).

Let T=\left\{t_\delta: \delta<\tau \right\} be a dense subspace of Y such that d(Y)=\lvert T \lvert=\tau. Note that \overline{T}=Y (closure is taken in Y). Let S=\bigcup_{\delta<\tau} S(t_\delta). Clearly \lvert S \lvert \le \tau. Consider the following subspace of X:

    X(S)=\left\{x \in X: S(x) \subset S  \right\}

It is clear that X(S) is a closed subspace of X. Since T \subset X(S), the closure of T (closure in X or in Y) is a subspace of X(S). Thus Y \subset X(S). Note that \overline{T}=Y \subset X(S). Since each X_\alpha has a countable base, the product space \prod_{\alpha<\tau} X_\alpha has a base of cardinality \tau. Thus \prod_{\alpha<\tau} X_\alpha has weight \le \tau. Since X(S) \subset \prod_{\alpha<\tau} X_\alpha, both Y and X(S) have weights \le \tau. We have w(Y) \le d(Y)=\tau. Note that d(Y) \le w(Y) always holds. Therefore d(Y)=w(Y). \blacksquare

Corollary 2
For any infinite cardinal \kappa, the \Sigma-product \Sigma(\kappa) is strongly monolithic.

Corollary 3
Any Corson compact space is strongly monolithic.

____________________________________________________________________

Blog posts on Corson compact spaces

____________________________________________________________________
\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