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

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!

• Should it be $\sigma$-product rather than $\Sigma$-product, since it means the countable-support product? This would align with $\sigma$-algebra, $\sigma$-additivity, and so forth. Apr 17 at 22:57
• This looks helpful: matstud.org.ua/ojs/index.php/matstud/article/view/274 Apr 17 at 23:40
• @JDH in this context, the lower-case $\sigma$ product means finite support, not countable. Apr 18 at 1:48
• @StevenClontz Thanks for that. But in truth, I find that usage very odd. Apr 18 at 2:03
• @JoelDavidHamkins You can blame H. H. Corson for the capital $\Sigma$ Apr 19 at 7:20

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.
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.