What is the extent of a $\Sigma$-product of a (uncountable) power of a (countable) discrete space?

Question

Recall that a $\Sigma$-product of a family of spaces $\{X_s:s\in S\}$ with a base point $a=(a_s)\in \prod_{s\in S} X_s$ is the subspace $$\Sigma(a)=\{x\in \prod_{s\in S} X_s: |\{s\in S:a_s\neq x_s\}|\leq\omega\}.$$

Several results about cardinal functions have been obtained for $\Sigma$-products in general by considering the cardinal function of each factor in the family. However, I have not be able to find any reference studying the extent of the $\Sigma$-product of a family of discrete spaces. To be more specific, what should be the extent of the $\Sigma$-product of powers of a discrete space? For example, what is the extent of $\Sigma(\mathbb{N}^{\omega_1})$?

Does anyone knows any reference where this question is investigated?

Thanks!

Answer 1

To answer the explicit question: the extent of every $\Sigma$-product of $\mathbb{N}$ is countable. H. H. Corson showed in Normality in subsets of product spaces, Amer. J. Math 81(1959), 785–796 that $\Sigma$-products of completely metrizable spaces are collectionwise normal. Hence their extent is bounded by their cellularity, which in the case of $\mathbb{N}$, or any Polish space, is countable.

By contrast the extent of the full product $\mathbb{N}^{\omega_1}$ is equal to $\aleph_1$. See Theorem A1.6 in Juhasz' Cardinal Functions in Topology.

Addendum (2024-04-19).

If a cardinal $\kappa$ carries the discrete topology then the cellularity of every power of $\kappa$ is equal to $\kappa$, for if $\mathcal{U}$ is a family of $\kappa^+$ many basic open sets then it can also be seen as a family of basic open sets on $\kappa^{\kappa^+}$. That power has density $\kappa$ (Hewitt-Marcewski-Pondiczery) and so $\mathcal{U}$ cannot be pairwise disjoint.

As above the $\Sigma$-product is collectionwise normal and as it is dense in the product its cellularity is equal to $\kappa$ and hence so is the extent.