Products of sequentially compact spaces versus $\mathfrak{t}$ and $\mathfrak{s}$

compactnessgeneral-topologyinfinitary-combinatoricsset-theory

Let $\mu$ be the minimal cardinal $\kappa$ such that there exists a sequence $(X_\alpha : \alpha < \kappa)$ with each $X_\alpha$ sequentially compact space such that $\prod_{\alpha < \kappa} X_\alpha$ is not sequentially compact.

Then, it is known that $\mathfrak{t} \le \mu \le \mathfrak{s}$.
Here, $\mathfrak{t}$ is the tower number and $\mathfrak{s}$ is the splitting number.
A proof is in E.K. van Douwen's article "The Integers and Topology" in "Handbook of Set-Theoretic Topology".

Now,

  1. can we separate $\mu$ from $\mathfrak{t}$ or separate $\mu$ from $\mathfrak{s}$?
    Or is it provable that $\mu = \mathfrak{t}$ or $\mu = \mathfrak{s}$ in ZFC?
  2. Since in the Mathias model $\mathfrak{t} = \aleph_1$ and $\mathfrak{s} = \aleph_2$, what can be said about $\mu$ in the model?

This question is the continuation of: For some sequentially compact space $X$, is $X^{\omega_1}$ not sequentially compact?

Best Answer

The number $\mu$ you are looking for is $\mathfrak h$. This is proved in

Petr Simon, “Products of sequentially compact spaces”, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 25 (1993), pp. 447-450.

The cardinals $\mathfrak t$, $\mathfrak h$ and $\mathfrak s$ are consistently distinct. In Mathias model $\mathfrak s = \mathfrak h$, see, e.g., the chapter by A. Blass in the Handbook of Set Theory.

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