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

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

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

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

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).

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.