# 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.

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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 $\kappa$.

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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.

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Blog posts on Corson compact spaces

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