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Alan Dow

Publications and source records attributed to Alan Dow.

At least 37 records · Page 2Linked to original sources
arXiv2022

S-spaces and large continuum

We prove that it is consistent with large values of the continuum that there are no S-spaces. We also show that we can also have that compact separable spaces of countable tightness have cardinality at most the continuum.

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arXiv2022

Spaces of countable free set number and PFA

The main result of this paper is that, under PFA, for every {\em regular} space $X$ with $F(X) = ω$ we have $|X| \le w(X)^ω$; in particular, $w(X) \le \mathfrak{c}$ implies $|X| \le \mathfrak{c}$. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. We also show that regularity cannot be weakened to Hausdorff in this result because we can find in ZFC a Hausdorff space $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. In fact, this space $X$ has the {\em strongly anti-Urysohn} (SAU) property that any two infinite closed sets in $X$ intersect, which is much stronger than $F(X) = ω$. Moreover, any non-empty open set in $X$ also has size $2^\mathfrak{c}$, and thus answers one of the main problems of \cite{JShSSz} by providing in ZFC a SAU space with no isolated points.

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arXiv2022

On the bounding, splitting, and distributivity numbers

The cardinal invariants $ \mathfrak h, \mathfrak b, \mathfrak s$ of $\mathcal P (ω)$ are known to satisfy that $ω_1 \leq \mathfrak h \leq\min\{\mathfrak b, \mathfrak s\}$. We prove that all inequalities can be strict. We also introduce a new upper bound for $\mathfrak h$ and show that it can be less than $\mathfrak s$. The key method is to utilize finite support matrix iterations of ccc posets following \cite{BlassShelah}.

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arXiv2021

On the weak pseudoradiality of CSC spaces

In this paper, we prove that in forcing extensions by a poset with finally property K over a model of GCH+$\square$, every compact sequentially compact space is weakly pseudoradial. We also prove the following assuming $\mathfrak{s}\leq \aleph_2$: (i) if $X$ is compact weakly pseudoradial, then $X$ is pseudoradial if and only if $X$ cannot be mapped onto $[0,1]^\mathfrak{s}$; (ii) if $X$ and $Y$ are compact pseudoradial spaces such that $X\times Y$ is weakly pseudoradial, then $X\times Y$ is pseudoradial. These results add to the wide variety of partial answers to the question by Gerlits and Nagy of whether the product of two compact pseudoradial spaces is pseudoradial.

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arXiv2021

Compact Spaces with a $P$-base

In the paper, we investigate (scattered) compact spaces with a $P$-base for some poset $P$. More specifically, we prove that, under the assumption $ω_1<\mathfrak{b}$, any compact space with an $ω^ω$-base is first-countable and any scattered compact space with an $ω^ω$-base is countable. These give positive solutions to Problems 8.6.9 and 8.7.7 in \cite{Banakh2019}. Using forcing, we also prove that in a model of $ω_1<\mathfrak{b}$, there is a non-first-countable compact space with a $P$-base for some poset $P$ with calibre~$ω_1$.

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arXiv2021

All Parovichenko spaces may be soft-Parovichenko

It is shown that, assuming the Continuum Hypothesis, compact Hausdorff space of weight at most $\mathfrak{c}$ is a remainder in a soft compactification of $\mathbb{N}$. We also exhibit an example of a compact space of weight $\aleph_1$ -- hence a remainder in some compactification of $\mathbb{N}$ -- for which it is consistent that is not the remainder in a soft compactification of $\mathbb{N}$.

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arXiv2020

On the cardinality of separable pseudoradial spaces

The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is, in ZFC, a regular (or just Hausdorff) SP of cardinality greater than $\mathfrak c$. While this question is left open, we establish a number of non-trivial results that we list: 1. It is consistent with Martin's Axiom and $\mathfrak c =\aleph_2$ that there is a countably tight and compact SP of cardinality $2^{\mathfrak c}$. 2. If $κ$ is a measurable cardinal then in the forcing extension obtained by adding $κ$ many Cohen reals, every countably tight regular SP space has cardinality at most $\mathfrak c$. 3. If $κ>\aleph_1$ Cohen reals are added to a model of GCH, then in the extension every pseudocompact SP space with a countable dense set of isolated points has cardinality at most $\mathfrak c$. 4. If $\mathfrak c\leq\aleph_2$, then there is a 0-dimensional SP space with a countable dense set of isolated points that has cardinal greater than $\mathfrak c$.

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arXiv2020

The independence of GCH and a combinatorial principle related to Banach-Mazur games

It was proved recently that Telgársky's conjecture, which concerns partial information strategies in the Banach-Mazur game, fails in models of $\mathsf{GCH}+\square$. The proof introduces a combinatorial principle that is shown to follow from $\mathsf{GCH}+\square$, namely: $\triangledown$: Every separative poset $\mathbb P$ with the $κ$-cc contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < κ$ for every $p \in \mathbb P$. We prove this principle is independent of $\mathsf{GCH}$ and $\mathsf{CH}$, in the sense that $\triangledown$ does not imply $\mathsf{CH}$, and $\mathsf{GCH}$ does not imply $\triangledown$ assuming the consistency of a huge cardinal. We also consider the more specific question of whether $\triangledown$ holds with $\mathbb P$ equal to the weight-$\aleph_ω$ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $\mathsf{ZFC}+\mathsf{GCH}$.

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arXiv2019

Telgarsky's conjecture may fail

Telgársky's conjecture states that for each $k \in \mathbb N$, there is a topological space $X_k$ such that in the Banach-Mazur game on $X_k$, the player {\scriptsize NONEMPTY} has a winning $(k+1)$-tactic but no winning $k$-tactic. We prove that this statement is consistently false. More specifically, we prove, assuming $\mathsf{GCH}+\square$, that if {\scriptsize NONEMPTY} has a winning strategy for the Banach-Mazur game on a $T_3$ space $X$, then she has a winning $2$-tactic. The proof uses a coding argument due to Galvin, whereby if $X$ has a $π$-base with certain nice properties, then {\scriptsize NONEMPTY} is able to encode, in each consecutive pair of her opponent's moves, all essential information about the play of the game before the current move. Our proof shows that under $\mathsf{GCH}+\square$, every $T_3$ space has a sufficiently nice $π$-base that enables this coding strategy. Translated into the language of partially ordered sets, what we really show is that $\mathsf{GCH}+\square$ implies the following statement, which is equivalent to the existence of the "nice'' $π$-bases mentioned above: \emph{Every separative poset $\mathbb P$ with the $κ$-cc contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < κ$ for every $p \in \mathbb P$.} We prove that this statement is independent of $\mathsf{ZFC}$: while it holds under $\mathsf{GCH}+\square$, it is false even for ccc posets if $\mathfrak{b} > \aleph_1$. We also show that if $|\mathbb P| < \aleph_ω$, then \axiom-for-$\mathbb P$ is a consequence of $\mathsf{GCH}$ holding below $|\mathbb P|$.

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arXiv2019

Pseudoradial spaces and copies of $ω_1+1$

In this paper we compare the concepts of pseudoradial spaces and the recently defined strongly pseudoradial spaces in the realm of compact spaces. We show that $\mathrm{MA}+\mathfrak{c}=ω_2$ implies that there is a compact pseudoradial space that is not strongly pseudoradial. We essentially construct a compact, sequentially compact space $X$ and a continuous function $f:X\toω_1+1$ in such a way that there is no copy of $ω_1+1$ in $X$ that maps cofinally under $f$. We also give some conditions that imply the existence of copies of $ω_1$ in spaces. In particular, $\mathrm{PFA}$ implies that compact almost radial spaces of radial character $ω_1$ contain many copies of $ω_1$.

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arXiv2019

Small cardinals and small Efimov spaces

We introduce and analyze a new cardinal characteristic of the continuum, the \emph{splitting number of the reals}, denoted $\mathfrak{s}(\mathbb R)$. This number is connected to Efimov's problem, which asks whether every infinite compact Hausdorff space must contain either a non-trivial convergent sequence, or else a copy of $β\mathbb N$.

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arXiv2018

Densely k-separable compacta are densely separable

A space has $σ$-compact tightness if the closures of $σ$-compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable space is separable. The somewhat surprising answer is that this property, for compact spaces, implies that every dense set is separable. The path to this result relies on the known connections established between $π$-weight and the density of all dense subsets, or more precisely, the cardinal invariant $δ(X)$.

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arXiv2018

Far points and discretely generated spaces

We give a partial solution to a question by Alas, Junqueria and Wilson by proving that under PFA the one-point compactification of a locally compact, discretely generated and countably tight space is also discretely generated. After this, we study the cardinal number given by the smallest possible character of remote and far sets of separable metrizable spaces. Finally, we prove that in some cases a countable space has far points.

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