The role of the index set in the product of uncountably many topological spaces 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.
 A: 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. 
A: The following is Example 11.8 of Hodel´s article in the Handbook of Set-Theoretic Topology, which provides several examples in the vein of Karol and Andreas's comment:
For the Cantor cube $X=2^\kappa$, we have:


*

*$\phi(X)=\kappa$ where $\phi$ is any of the cardinal functions: spread, hereditary Lindelof degree, hereditary density, net weight, weight, $\pi$-weight, hereditary $\pi$-weight, point separating weight, diagonal degree, tightness, $\pi$-character, hereditary $\pi$-character, pseudo-character, character.

*$d(X)=\log \kappa$ (this is Karol-Andreas's example).

