Alan Dow

Densely k-separable compacta are densely separable

Alan Dow, Istvan Juhasz

A space has $I ¨ $ -compact tightness if the closures of $I ¨ $ -compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider th…

On the weak pseudoradiality of CSC spaces

Hector Barrig-Acosta, Alan Dow

In this paper, we prove that in forcing extensions by a poset with finally property K over a model of GCH+ $□$ , every compact sequentially compact space is weakly pseudoradial…

Weight, net weight, and elementary submodels

Alan Dow, IstvÃ¡n JuhÃ¡sz

In this note we prove several theorems that are related to some results and problems from [6]. We answer two of the main problems that were raised in [6]. First we give a ZFC examp…

math.LO2007

More on Tie-points and homeomorphism in N^*

Alan Dow, Saharon Shelah

A point x is a (bow) tie-point of a space X if X setminus {x} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construct…

The Category Dichotomy for Ideals

Alan Dow, Raul Figueroa-Sierra, Osvaldo GuzmÃ¡n +1

We provide a counterexample to the Category Dichotomy in the framework of $ZFC$ . That is, we prove the existence of an ideal on $I ¨ $ that is not KatÄtov below $\mathsf{nw…

math.GN2019

Pseudoradial spaces and copies of $I ¨ _{1} + 1$

Angelo Bella, Alan Dow, Rodrigo HernÃ¡ndez-GutiÃ©rrez

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}+\mathfr…

Not OCA and products of Frechet spaces

Alan Dow

We continue the investigation of the question of whether the product of two countable FrÃ©chet spaces must be M-separable. We are especially interested in this question in the pres…

math.LO1993

Äech-Stone remainders of spaces that look like $[0, \infty)$

Alan Dow, Klaas Pieter Hart

We show that many spaces that look like the half~line~$\halfline=[0,\infty)$ have, under~$\CH$, a Äech-Stone-remainder that is homeomorphic to~$\Hstar$. We also show that $\CH$ is…

$PFA$ and $I ¨ _{1}$ -free compact spaces

Alan Dow, Klaas Pieter Hart

The Proper Forcing Axiom implies that compact Hausdorff spaces are either first-countable or contain a converging $I ¨ _{1}$ -sequence.

math.GN2022

Spaces of countable free set number and PFA

Alan Dow, Istvan Juhasz

The main result of this paper is that, under PFA, for every {\em regular} space $X$ with $F (X) = I ¨ $ we have $∣ X ∣ \leq w (X)^{I ¨} $ ; in particular, $w (X) \leq c$ implies $|X|…

math.GN2007

A separable non-remainder of H

Alan Dow, Klaas Pieter Hart

We prove that there is a compact separable continuum that (consistently) is not a remainder of the real line.

math.GN2024

Autohomeomorphisms of pre-images of $N^{*}$

Alan Dow

In the study of the Stone-uCech remainder of the real line a detailed study of the Stone-uCech remainder of the space $N \times [0, 1]$ , which we denote as $M$ , has o…

math.LO2022

S-spaces and large continuum

Alan Dow, Saharon Shelah

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 tightnes…

All spaces of countable spread can be small

Alan Dow, IstvÃ¡n JuhÃ¡sz

The main result of this paper is the proof of the simultaneous consistency, modulo a weakly compact cardinal, of the equality $2^{< c} = c$ with the following…

No covering with nowhere dense \textsf{P}-sets in the Cohen model

Alan Dow, Osvaldo GuzmÃ¡n

We prove that if less than $ℵ_{I ¨} $ -many Cohen reals are added to a model of \textsf{CH}, then $I ¨ ^{*}$ can not be covered by nowhere dense \textsf{P}-sets (equivalently, the…

math.GN2015

The Katowice problem and autohomeomorphisms of $I ¨ ^{*}$

David Chodounsky, Alan Dow, Klaas Pieter Hart +1

We show that the existence of a homeomorphism between $I ¨ _{0}^{*}$ and $I ¨ _{1}^{*}$ entails the existence of a non-trivial autohomeomorphism of $I ¨ _{0}^{*}$ .

math.GN2023

Closed copies of $N$ in $R^{I ¨ _{1}}$

Alan Dow, Klaas Pieter Hart, Jan van Mill +1

We investigate closed copies of~ $N$ in powers of~ $R$ with respect to $C^{*}$ - and $C$ -embedding. We show that $R^{I ¨ _{1}}$ contains closed copies of~$\mathbb…

math.LO2000

The measure algebra does not always embed

Alan Dow, Klaas Pieter Hart

The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(N)/fin. We also discuss the errors in previous results on the embeddability of the measure algeb…

Far points and discretely generated spaces

Alan Dow, Rodrigo HernÃ¡ndez-GutiÃ©rrez

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 count…

Ultrafilters in the random real model

Alan Dow, Osvaldo GuzmÃ¡n

We prove that \textsf{P}-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $ℵ_{I ¨} $ -many random reals to a mode…

math.GN2023

Some realcompact spaces

Alan Dow, Jan van Mill, Klaas Pieter Hart +1

We present examples of realcompact spaces with closed subsets that are C*-embedded but not C-embedded, including one where the closed set is a copy of the space of natural numbers.

Reversible filters

Alan Dow, Rodrigo HernÃ¡ndez-GutiÃ©rrez

A space is reversible if every continuous bijection of the space onto itself is a homeomorphism. In this paper we study the question of which countable spaces with a unique non-iso…

math.GN2000

Hereditary indecomposability and the Intermediate Value Theorem

Alan Dow, Klaas Pieter Hart

We show that hereditarily indecomposable spaces can be characterized by a special instance of the Intermediate Value Theorem in their rings of continuous functions.

math.GN2020

Productivity of cellular-Lindelof spaces

Alan Dow, Robert M. Stephenson

The main purpose of this note is to prove that the product of a cellular Lindelof space with a space of countable spread need not be cellular-Lindelof.

$M^{}$ , $N^{}$ , and $H^{*}$

Will Brian, Alan Dow, Klaas Pieter Hart

Let $M = N \times [0, 1]$ . The natural projection $I ¨ : M \to N$ , which sends $(n, x)$ to $n$ , induces a projection mapping $Ï^*: \mathbb{M}…

Compact Spaces with a $P$ -base

Alan Dow, Ziqin Feng

In the paper, we investigate (scattered) compact spaces with a $P$ -base for some poset $P$ . More specifically, we prove that, under the assumption $I ¨ _{1} < b$ , any compact…

math.LO2018

On the cofinality of the splitting number

Alan Dow, Saharon Shelah

The splitting number can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and the second author "…

All Parovichenko spaces may be soft-Parovichenko

Alan Dow, Klaas Pieter Hart

It is shown that, assuming the Continuum Hypothesis, compact Hausdorff space of weight at most $c$ is a remainder in a soft compactification of $N$ . We also exh…

Normality versus paracompactness in locally compact spaces

Alan Dow, Franklin D. Tall

This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing…

math.GN2026

A parametrized $♢$ for the Laver property and nontrivial automorphisms of $P (I ¨ ) / Fin$

Will Brian, Alan Dow

We introduce a new parametrized diamond principle denoted $♢ (LP)$ . This principle is akin to the parametrized diamonds of Moore, HruÅ¡Ã¡k, and DÅ¾amonja, each o…

On the tightness of $G_{I^} ´$ -modifications

Alan Dow, IstvÃ¡n JuhÃ¡sz, Lajos Soukup +2

The $G_{I^} ´$ -modification $X_{I^} ´$ of a topological space $X$ is the space on the same underlying set generated by, i.e. having as a basis, the collection of all $G_{I^} ´$ subsets of $X$ .…

math.GN2014

On subcontinua and continuous images of beta R\R

Alan Dow, Klaas Pieter Hart

We prove that the Cech-Stone remainder of the real line has a family of 2^c mutually non-homeomorphic subcontinua. We also exhibit a consistent example of a first-countable continu…

Pseudo P-points and splitting number

Alan Dow, Saharon Shelah

We construct a model in which the splitting number is large and every ultrafilter has a small subset with no pseudo-intersection.

Some results on the $I ¨ $ -weight of countable FrÃ©chet-Urysohn spaces

Alan Dow

The $I ¨ $ -weight spectrum for countable regular FrÃ©chet-Urysohn spaces is the set of uncountable cardinals that are equal to the $I ¨ $ -weight for some such space. We determine this…

math.GN2013

Reflecting LindelÃ¶f and converging omega_1-sequences

Alan Dow, Klaas Pieter Hart

We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging omega-sequence or a non-trivial converging omega_1-sequence. We…

Hereditarily normal manifolds of dimension > 1 may all be metrizable

Alan Dow, Franklin D. Tall

P.J. Nyikos has asked whether it is consistent that every hereditarily normal manifold of dimension > 1 is metrizable, and proved it is if one assumes the consistency of a supercom…

math.LO2007

Tie-points and fixed-points in N^*

Alan Dow, Saharon Shelah

A point x is a (bow) tie-point of a space X if X setminus {x} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construct…

math.GN2020

On the cardinality of separable pseudoradial spaces

Alan Dow, Istvan Juhasz

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 her…

math.GN2014

CH and the Moore-Mrowka Problem

Alan Dow, Todd Eisworth

We show that the Continuum Hypothesis is consistent with all regular spaces of hereditarily countable $I ¨ $ -character being C-closed. This gives us a model of ZFC in which the Conti…

math.GN2000

Applications of another characterization of betaN\N

Alan Dow, Klaas Pieter Hart

Steprans provided a characterization of betaN\N in the aleph_2-Cohen model that is much in the spirit of Parovicenko's characterization of this space under CH. A variety of the top…

math.GN2005

Cosmic dimensions

Alan Dow, Klaas Pieter hart

Martin's Axiom for $I ¨ $ -centered partial orders implies that there is a cosmic space with non-coinciding dimensions.

math.LO2020

An Efimov space with character less than $s$

Alan Dow

It is consistent that there is a compact space of character less than the splitting number in which there are no converging sequences. Such a space is an Efimov space.

math.GN2024

$C$ -embedding, LindelÃ¶fness, and Äech-completeness

Alan Dow, Klaas Pieter Hart, Jan van Mill +1

We show that in the class of LindelÃ¶f Äech-complete spaces the property of being $C$ -embedded is quite well-behaved. It admits a useful characterization that can be used to show…

Compact spaces with a $P$ -diagonal

Alan Dow, Klaas Pieter Hart

We prove that compact Hausdorff spaces with a $P$ -diagonal are metrizable.

math.LO2019

Telgarsky's conjecture may fail

Will Brian, Alan Dow, David Milovich +1

TelgÃ¡rsky's conjecture states that for each $k \in N$ , there is a topological space $X_{k}$ such that in the Banach-Mazur game on $X_{k}$ , the player {\scriptsize NONEMPTY} ha…

math.LO2022

Automorphisms of $\mathcal P(Ï)/\mbox{fin}$ and large continuum

Alan Dow

We prove that it is consistent with $c > ℵ_{2}$ that all automorphisms of $\mathcal P(Ï)/\mbox{fin}$ are trivial.

math.GN2011

Elementary chains and compact spaces with a small diagonal

Alan Dow, Klaas Pieter Hart

It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary c…

math.GN2000

A Universal Continuum of Weight aleph

Alan Dow, Klaas Pieter Hart

We prove that every continuum of weight aleph_1 is a continuous image of the Cech-Stone-remainder R^* of the real line. It follows that under CH the remainder of the half line [0,i…

math.LO2026

Nontrivial automorphisms of $P (I ¨ ) / Fin$ in Cohen models

Will Brian, Alan Dow

We show that if $I^º < ℵ_{I ¨} $ Cohen reals are added to a model of $CH$ , then there are nontrivial automorphisms of $P (I ¨ ) / Fin$ in the extension. Under…

math.LO2020

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

Will Brian, Alan Dow, Saharon Shelah

It was proved recently that TelgÃ¡rsky's conjecture, which concerns partial information strategies in the Banach-Mazur game, fails in models of $GCH + □$ . The proof in…

A zero-dimensional F-space that is not strongly zero-dimensional

Alan Dow, Klaas Pieter Hart

We present an example of a zero-dimensional $F$ -space that is not strongly zero-dimensional.

math.GN2024

Non-trivial copies of N*

Alan Dow

We show that it is consistent to have regular closed non-clopen copies of $N^{*}$ within $N^{*}$ and a non-trivial self-map of $N^{*}$ even if all autohomeomorphi…

math.GN2023

On Roitman's principles $MH$ and $I^$

Hector Barriga-Acosta, Will Brian, Alan Dow

The Model Hypothesis (abbreviated $MH$ ) and $I^$ are set-theoretic axioms introduced by J. Roitman in her work on the box product problem. Answering some questions of Roit…

Asymmetric tie-points and almost clopen subsets of N*

Alan Dow, Saharon Shelah

A tie-point of a compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. Set-theoretically a tie-point of…

New examples in the study of selectively separable spaces

Alan Dow, Hayden Pecoraro

The property of being selectively separable is well-studied and generalizations such as H-separable and wH-separable have also generated much interest. Bardyla, Maesano, and Zdomsk…

math.GN2008

A first-countable non-remainder of H

Alan Dow, Klaas Pieter Hart

We give a (consistent) example of a first-countable continuum that is not a remainder of the real line.

math.LO2019

Small cardinals and small Efimov spaces

Will Brian, Alan Dow

We introduce and analyze a new cardinal characteristic of the continuum, the \emph{splitting number of the reals}, denoted $s (R)$ . This number is connected to Ef…

math.LO2022

On the bounding, splitting, and distributivity numbers

Alan Dow, Saharon Shelah

The cardinal invariants $h, b, s$ of $P (I ¨ )$ are known to satisfy that $I ¨ _{1} \leq h \leq min {b, s}$ . We pro…

math.LO2016

PFA(S)[S] and countably compact spaces

Alan Dow, Franklin D. Tall

We show a number of undecidable assertions concerning countably compact spaces hold under PFA(S)[S]. We also show the consistency without large cardinals of "every locally compact,…