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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.

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  • $\begingroup$ Well, if $X$ is any separable space with at least 2 points, then $\prod_IX$ is separable iff $I$ is countable - does this count? $\endgroup$ – Noah Schweber Nov 11 '15 at 1:56
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    $\begingroup$ @Noah, the product of $\mathfrak{c}$ many separable spaces is separable. That's Hewitt–Marczewski–Pondiczery theorem. $\endgroup$ – Karol Szumiło Nov 11 '15 at 2:03
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    $\begingroup$ 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. $\endgroup$ – David Handelman Nov 11 '15 at 3:26
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    $\begingroup$ 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. $\endgroup$ – dalry Nov 11 '15 at 6:19
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    $\begingroup$ 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 $\mathfrak c^+$. $\endgroup$ – Andreas Blass Nov 11 '15 at 11:49
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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.

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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).
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