TITLE: The role of the index set in the product of uncountably many topological spaces
QUESTION [6 upvotes]: Let $‎\langle‎ ‎‎X_i,\mathcal{T}_i \rangle_{i\in I}$ be a family of topological spaces. Consider $X=\prod_{i\in I} X_i$ with product topology.
Question. Is there a topological property that holds in $X$ when $|I|=\aleph_{\beta}$, but does not hold whenever $|I|=\aleph_{\alpha}$, for some $0<\alpha<\beta$ ?
Any reference or helpful comment will be appreciated.

REPLY [2 votes]: The splitting number $\mathfrak{s}$ is the minimal cardinal $\kappa$ such that if $\mathcal{A} \subseteq \mathcal{P}(\omega)$ has size $< \kappa$, there exists some $X \subseteq \omega$ such that for all $A \in \mathcal{A}$, $X \setminus A$ is finite or $X \cap A$ is finite.
Then if all $X_i$ are sequentially compact Tychonoff spaces, and $|I| <\mathfrak{s}$ then $\prod_{i \in I} X_i$ is also sequentially compact. And $\{0,1\}^{\mathfrak{s}}$ is not sequentially compact. What $\aleph_\alpha$ equals $\mathfrak{s}$ cannot be said in ZFC. It could be $\alpha = 1$ (under CH, e.g.) or much bigger, depending on the size of the continuum. See van Douwen's paper in the Handbook of Set Theoretic Topology on cardinal invariants of the continuum.