I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality.

Searching in several textbooks and articles, seems that there is no non-trivial application of this result yet (i.e. all the applications (until now) that I have found are obtained just changing the cardinals: so, if $\mathfrak{p}$ has some topological property, then $\mathfrak{t}$ also has it).

So, my question is: are there any interesting consequence of $\mathfrak{p}=\mathfrak{t}$ in Topology?

  • 2
    $\begingroup$ The paper (pdfs.semanticscholar.org/d7c3/… of Alas and Wilson contains some interesting topological applications of the small uncountable cardinals $\mathfrak p$ and $\mathfrak t$ and those applications differ for those (eventually equal) cardinals $\mathfrak p=\mathfrak t$. $\endgroup$ – Taras Banakh Jan 17 at 12:02
  • $\begingroup$ It's an interesting paper, but mybe my question wasn't clear enough. What I want to know is if somebody has found an application like this: "if $\mathfrak{p}=\mathfrak{t}$, then (something)". There are several topological applications derivated from the equality of a pair of these cardinals, e.g. Kojman-Shelah showed that $\mathfrak{p}=\mathfrak{c}$ implies the existence of a van der Waerden space that is not a Hindman space. These are the applications I'm trying to find $\endgroup$ – Alexei0709 Jan 21 at 19:13

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