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Is there a consistent example of a compact Hausdorff space $X$ on which the following holds?

i) there is a $Y \in {[X]}^{\aleph_1}$ such that $\psi (Y) = \aleph_1$; and

ii) there is no non-trivial converging sequence of type $\omega_1$ in $X$.

Note that the negation of (ii) implies (i) (assuming ZFC of course).

I know only one example of a compact Hausdorff space $X$, with $\psi (X) \geq \aleph_1$, satisfying (ii). It is the one-point compactification $X$ of the (very complex) locally compact space $Y$ constructed in

I. Juhász, P. Koszmider and L. Soukup, A first countable, initially $\omega_{1}$-compact but non-compact space, Topology and its Applications 156 (2009), 1863-1879. doi:10.1016/j.topol.2009.04.004

In this case, I don't know if $X$ satisfies (i).

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    $\begingroup$ By "a converging sequence of type $\omega_1$", do you mean a copy of $\omega_1$ with the order topology, converging to some $x$ in the sense that any neighborhood of $x$ contains a terminal part ? $\endgroup$
    – Mathieu Baillif
    Commented Aug 29, 2013 at 20:18
  • $\begingroup$ Essentially, yes. But not necessarily with the order topology in the whole sequence. The important thing to me here is: "any neighborhood of $x$ contains a terminal part". $\endgroup$
    – Alberto Levi
    Commented Aug 29, 2013 at 20:52
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    $\begingroup$ What is $\psi$? $\endgroup$
    – Joel David Hamkins
    Commented Dec 8, 2014 at 1:42
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    $\begingroup$ $\psi$ is the cardinal function pseudo-character: $\psi (X)$ is the smallest infinite cardinal $\kappa$ such that each point of $X$ is the intersection of a family of cardinality $\leq \kappa$ of sets which are open in $X$. $\endgroup$
    – Alberto Levi
    Commented Dec 8, 2014 at 1:48
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    $\begingroup$ What is ${[X]}^{\aleph_1}$? A family of closed subspaces of density $\aleph_1$? $\endgroup$
    – Adam Przeździecki
    Commented Dec 10, 2014 at 9:41

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