# Well-founded families of sets and topological convergence

## Background/Motivation

A space is scattered if every non-empty subset has an isolated point. A space is pseudoradial if every non-closed set contains a transfinite sequence (a well-ordered net) converging outside of it. A classical result of Mrowka, Rajagopalan and Soundararajan from "A characterization of compact scattered spaces through chain limits (chain compact spaces)" provides a surprising link between these two notions.

Every compact scattered space is pseudoradial.

A recurring theme in set-theoretic topology is the extension of results about compact spaces to Lindelof $P$-spaces (that is, Lindelof spaces where $G_\delta$ sets are open). This is because Lindelof $P$-spaces share with compact spaces many of the properties that are usually lost in Lindelof spaces (finite productivity, normality, etc...). There are several results of this kind scattered in the literature, and I think that Juhasz, Soukup, Szentmiklossy and Weiss are working on a systematic study, at least as far as cardinal invariants on Lindelof $P$-spaces are concerned.

## Three questions in increasing order of strength

QUESTION 1: Is there a Lindelof scattered P-space which is not pseudoradial?

The only partial result to the above question I have so far is that every Lindelof scattered P-space of pseudocharacter less than $\aleph_\omega$ is pseudoradial (the pseudocharacter at a point $x$ is defined as the minimum $\kappa$ such that $\{x\}$ is a $G_\kappa$ set, and the pseudocharacter of the space is obtained simply by taking the supremum over all points). See my paper with Bella and Costantini "P-spaces and the Whyburn property", Theorem 19.

The following question suggests a possible path to a counterexample.

QUESTION 2: Is there a compact scattered $X$ space with a point $p$ of uncountable character such that no sequence of uncountable regular length converges to $p$?

Here is why. The $G_\delta$-modification of a topological space $X$ is simply the topology generated by all countable intersections of open sets of $X$. Since the $G_\delta$ modification of compact scattered spaces is Lindelof (see Arhangel'skii's function spaces book), the $G_\delta$ modification of a space positively answering QUESTION 2 would be a Lindelof scattered $P$-space where no non-trivial sequence converges to $p$.

I wonder if to answer Question 2 one could use a general construction of locally compact scattered spaces described in Juhasz, Shelah, Soukup and Szentmiklossy's paper "A tall space with a small bottom"

Let $A$ be a family of sets, and consider the coarsest topology $\tau_A$ on $A$ making $P(a) \cap A$ clopen, for every $a \in A$. Call a family of sets $A$ well-founded, if the containment relation on $A$ is well-founded. Call a family of sets $A$ $\cap$-closed if $a \cap b \in A \cup \{\emptyset\}$ for every $a, b \in A$. Well-foundedness of $A$ guarantees that the resulting space $(A, \tau_A)$ is scattered, and along with $\cap$-closedness it makes each $P(a) \cap A$ compact, so that $(A, \tau_A)$ is also locally compact. To state my final question concisely I need the following definitions.

Call a family of sets $A$ thick if for every subfamily $E \subset A$ having regular uncountable cardinality $\kappa$, there is $a \in A$ such that $P(a) \cap E$ has cardinality $\kappa$, where $P(a)$ is the powerset of a.

Call a family of sets $A$ small if there is a countable subfamily $C \subset A$ such that $\bigcup \{ P(c) \cap A: c \in C \}=A$.

QUESTION 3: Is there a well-founded $\cap$-closed family of sets $A$, which is thick but not small?

Thickness serves to guarantee no transfinite uncountable sequence of regular length converges to the added point in the one-point compactification of the corresponding locally compact scattered space, and avoiding smallness serves to guarantee the added point in the one-point compactification of $(A, \tau_A)$ does not have countable character.

Summing it all up, a yes to Question 3 implies a yes to Question 2 (just take the one point compactification of the resulting locally compact scattered space $(A, \tau_A)$), which in turn implies a yes to Question 1. Not every locally compact scattered space arises in the way described above, so there may well be other ways to answer Question 2 in the positive. And there may well be other ways to answer Question 1 as well.

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