# 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.

• Well, if $X$ is any separable space with at least 2 points, then $\prod_IX$ is separable iff $I$ is countable - does this count? – Noah Schweber Nov 11 '15 at 1:56
• @Noah, the product of $\mathfrak{c}$ many separable spaces is separable. That's Hewitt–Marczewski–Pondiczery theorem. – Karol Szumiło Nov 11 '15 at 2:03
• If $X_i$ are all metrizable and have at least two points, then the property, not metrizable, holds for $\beta =\aleph_1$, but not for any cardinal strictly less. I doubt this is what you had in mind. – David Handelman Nov 11 '15 at 3:26
• I think that the concept of cardinal functions in topology might provide examples of what you are looking for. In the eponymous book by Juhász (readily available online) there is a whole chapter devoted to the topic of their behaviour with respect to products. – dalry Nov 11 '15 at 6:19
• Addendum to the comment by @KarolSzumiło : The product of $\mathfrak c^+$ copies of a two-point discrete space is not separable. So separability is an example of what the OP asked about, with the "break" occurring at the step from $\mathfrak c$ to $\mathfrak c^+$. – Andreas Blass Nov 11 '15 at 11:49

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.
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).