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!