Convergence properties in dense subsets of $\omega^*$

Question

The space $\omega^*$, the remainder of the Cech-Stone compactification of the integers, fails to have all convergence-type properties known to me.

1. Sequentiality. (As a matter of fact $\omega^*$ does not even have any convergence sequences, because every infinite closed subset of $\omega^*$ has cardinality $2^\mathfrak{c}$).

2. Pseudoradiality. (Every infinite compact pseudoradial space must contain a convergent sequence).

3. Countable tightness (As proved by Kunen, $\omega^*$ even contains points which are not accumulation points of every countable subset of $\omega^*$).

4. The "weak Whyburn property" (that means, for every non-closed $A \subset X$ there is $x \in \overline{A} \setminus A$ and $B \subset A$ such that $\overline{B} \setminus B=\{x\}$. Note that every compact weakly Whyburn space must contain a convergent sequence).

5. Discrete generability. That follows from Proposition 2.4 of:

Tkachuk, V. V.; Wilson, R. G., Box products are often discretely generated, Topology Appl. 159, No. 1, 272-278 (2012). ZBL1236.54005.

or from the fact that $\omega^*$ contains a point which is in an accumulation point of a countable set, but not an accumulation point of any countable discrete set (see Theorem 4.4.1 in Jan van Mill's article in the Handbook of set-theoretic topology).

Obviously a dense subset of $\omega^*$ cannot be sequential, but:

QUESTION: Can a dense subset of $\omega^*$ be pseudoradial, countably tight, weakly Whyburn or discretely generated?

Parts of the above question have already been asked in the literature and in the comment section of a question on Mathoverflow (I'll give references below), but I thought it would be nice to have them all in one place.

• Is there a compact space with no countably generated dense subspace? (see the comment section of my answer)
• Alas, Ofelia T.; Madriz-Mendoza, Maira; Wilson, Richard G., Some results and examples concerning Whyburn spaces, Appl. Gen. Topol. 13, No. 1, 11-19 (2012); corrigendum ibid. No. 2, 225-226 (2012). ZBL1245.54024. (Question 4.4 even asks whether $\omega^*$ can have a dense Whyburn subspace).

EDIT (22/07/2021): Let me remark that under CH $\omega^*$ has a dense subspace which is radial, Whyburn and discretely generated. It suffices to take the set $D$ of all $P$-points.

Answer 1

I have been asked to add my comment as an answer, so here it is. A dense subset of $\omega^*$ cannot be countably tight. The reason is that if $x$ is any element of $\omega^*$, there is an open subset $U$ of $\omega^*$ whose closure contains $x$ but such that $x$ is not in the closure of any countable subset of $U$, and this property carries over to dense subsets.

To get $U$ consider two cases. Case 1. If $x$ is a P-point, let $U = \omega^* \setminus \{x\}$. Case 2. If $x$ is not a P-point, it is on the boundary of a zero-set $Z$; let $U = Int_{\omega^*}Z$. Since zero-sets in $\omega^*$ have dense interiors, $x$ is in the closure of $U$. It follows from the fact that $\omega^*$ is an F-space that the $\omega^*$-closure of every countable subset of $U$ is a subset of $U$, and, therefore, does not contain $x$.

Answer 2

Comments from Alan Dow:

(1) under CH the set of P-points is dense and radial even, using $\omega_1$-sequences.

(2) if $D$ is dense and pseudoradial then for every cozero set $C$ of $\omega^*$ the intersection $C\cap D$ is closed in $D$. Say $C=f^{-1}[(0,1]]$; a convergent sequence $\langle x_\alpha:\alpha<\kappa\rangle$ in $C\cap D$ should have uncountable cofinality, but then there is an $n$ such that $\{\alpha:f(x_\alpha)\ge2^{-n}\}$ is unbounded and hence the limit wound be in $\{x:f(x)\ge2^{-n}\}$.

(3) if there are no P-points then every point of the dense set $D$ is in the boundary of some cozero set $C$, for that $C$ the intersection $C\cap D$ is not closed; hence $D$ is not pseudoradial.