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Will Brian

Publications and source records attributed to Will Brian.

At least 19 recordsLinked to original sources
arXiv2026

Nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$ in Cohen models

We show that if $\kappa < \aleph_\omega$ Cohen reals are added to a model of $\mathsf{CH}$, then there are nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$ in the extension. Under some further hypotheses on the ground model, namely the existence of long enough sage Davies trees (which follows from $\mathsf{SCH}$ plus $\square_\lambda$ for every $\lambda$ with $\mathrm{cf}(\lambda) = \omega$), we prove the same result for cardinals $\kappa \geq \aleph_\omega$ as well. This extends a result a Shelah and Stepr\={a}ns, who proved the result for $\kappa = \aleph_2$.

math.LO↗
arXiv2026

A parametrized $\diamondsuit$ for the Laver property and nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$

We introduce a new parametrized diamond principle denoted $\diamondsuit(\mathsf{LP})$. This principle is akin to the parametrized diamonds of Moore, Hru\v{s}\'ak, and D\v{z}amonja, each of which corresponds to some cardinal invariant of the continuum, and gives a $\diamondsuit$-like guessing principle implying the corresponding invariant is $\aleph_1$. Our principle $\diamondsuit(\mathsf{LP})$ is a $\diamondsuit$-like guessing principle implying the Laver property holds over a given inner model, such as the ground model in a forcing extension. We show $\diamondsuit(\mathsf{LP})$ holds in many familiar models of $\mathsf{ZFC}$ obtained by forcing, namely those obtained from a model of $\mathsf{CH}$ by a length-$\omega_2$ countable support iteration of proper Borel posets with the Laver property. This is true for essentially the same reason that the usual parametrized diamonds hold in similarly described forcing extensions where their corresponding cardinal invariant is $\aleph_1$. We also prove that if $\diamondsuit(\mathsf{LP})$ holds over an inner model of $\mathsf{CH}$ then there are nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$; in fact we get particularly nice automorphisms extending nontrivial involutions built around $P$-points in the ground model. Additionally, we show that, like the Sacks model, all automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$ are somewhere trivial in the Mathias model. This puts a limitation on the kinds of automorphisms obtainable from $\diamondsuit(\mathsf{LP})$.

math.GN↗
arXiv2025

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

Let $\mathbb{M} = \mathbb N \times [0,1]$. The natural projection $\pi: \mathbb{M} \rightarrow \mathbb N$, which sends $(n,x)$ to $n$, induces a projection mapping $\pi^*: \mathbb{M}^* \rightarrow \mathbb N^*$, where $\mathbb{M}^*$ and $\mathbb N^*$ denote the \v{C}ech-Stone remainders of $\mathbb{M}$ and $\mathbb N$, respectively. We show that $\mathsf{CH}$ implies every autohomeomorphism of $\mathbb N^*$ lifts through the natural projection to an autohomeomorphism of $\mathbb{M}^*$. That is, for every homeomorphism $h: \mathbb N^* \rightarrow \mathbb N^*$ there is a homeomorphism $H: \mathbb{M}^* \rightarrow \mathbb{M}^*$ such that $\pi^* \circ H = h \circ \pi^*$. This complements a recent result of the second author, who showed that this lifting property is not a consequence of $\mathsf{ZFC}$. Combining this lifting theorem with a recent result of the first author, we also prove that $\mathsf{CH}$ implies there is an order-reversing autohomeomorphism of~$\mathbb H^*$, the \v{C}ech-Stone remainder of the half line $\mathbb H = [0,\infty)$.

math.GN↗
arXiv2024

Small-dimensional normed barrelled spaces

We prove that every separable Banach space has a barrelled subspace with algebraic dimension $\mathrm{non}(\mathcal M)$, which denotes the smallest cardinality of a non-meager subset of $\mathbb R$. This strengthens a theorem of Sobota. More generally, we prove that every Banach space with density character $\kappa$ contains a barrelled subspace with algebraic dimension $\mathrm{cf}[\kappa]^\omega \cdot \mathrm{non}(\mathcal M)$, and in particular it is consistent with $\mathsf{ZFC}$ that every Banach space with density character $<\!\mathfrak{c}$ has a barrelled subspace with dimension $<\!\mathfrak{c}$. We also prove that if the dual of a Banach space contains either $c_0$ or $\ell^p$ for some $p \geq 1$, then that space does not have a barrelled subspace with dimension $<\!\mathrm{cov}(\mathcal N)$, which denotes the smallest cardinality of a collection of Lebesgue null sets covering $\mathbb R$. In particular, it is consistent with $\mathsf{ZFC}$ that no classical Banach spaces contain barrelled subspaces with dimension $\mathfrak{b}$. This partly answers a question of S\'anchez Ruiz and Saxon.

math.FA↗
arXiv2024

Conjugating trivial automorphisms of $\mathcal P(\mathbb N)/\mathrm{Fin}$

A trivial automorphism of the Boolean algebra $\mathcal P(\mathbb N) / \mathrm{Fin}$ is an automorphism induced by the action of some function $\mathbb N \rightarrow \mathbb N$. In models of forcing axioms all automorphisms are trivial, and therefore two trivial automorphisms are conjugate if and only if they have the same (modulo finite) cycle structure. We show that the Continuum Hypothesis implies that two trivial automorphisms are conjugate if and only if there are neither first-order obstructions nor index obstructions for their conjugacy. This is equivalent to given trivial automorphisms being conugate in some forcing extension of the universe. To each automorphism $\alpha$ of $\mathcal P(\mathbb N) / \mathrm{Fin}$ we associate the first-order structure $\mathfrak{A}_\alpha=(\mathcal P(\mathbb N) / \mathrm{Fin},\alpha)$ and compute the existential theories of these structures. These results are applied to resolve a question of Braga, Farah, and Vignati and prove that there are coarse metric spaces $X$ and $Y$ such that the isomorphism between their uniform Roe coronas is independent from $\mathsf{ZFC}$.

math.LO↗
arXiv2024

Endpoint-homogeneous fans

A fan $F$ is \emph{endpoint-homogeneous} if for any two endpoints $e,e'$ of $F$, there is a homeomorphism $h: F \rightarrow F$ such that $h(e) = e'$. We prove there are uncountably many distinct homeomorphism types of endpoint-homogeneous smooth fans. To do this, we associate to each such fan $F$ a topological invariant, in the form of a characteristic subset $EPG(F) \subseteq [0,1]$ describing how the endpoints of $F$ limit onto any given blade of $F$. We describe precisely all the uncountably many different $X \subseteq [0,1]$ that can arise as $EPG(F)$ for some endpoint-homogeneous smooth fan $F$. We also prove the existence of $\frac{1}{n}$-homogeneous smooth fans for all $n \geq 5$.

math.GN↗
arXiv2024

Does $\mathcal P(\omega) / \mathrm{fin}$ know its right hand from its left?

Let $\sigma$ denote the shift automorphism on $\mathcal{P}(\omega) / \mathrm{fin}$, defined by setting $\sigma([A]) = [A+1]$ for all $A \subseteq \omega$. We show that the Continuum Hypothesis implies the shift automorphism $\sigma$ and its inverse $\sigma^{-1}$ are conjugate in the automorphism group of $\mathcal{P}(\omega) / \mathrm{fin}$. Due to work of van Douwen and Shelah, it has been known since the 1980's that it is consistent with $\mathsf{ZFC}$ that $\sigma$ and $\sigma^{-1}$ are not conjugate. Our result shows that the question of whether $\sigma$ and $\sigma^{-1}$ are conjugate is independent of $\mathsf{ZFC}$. As a corollary to the main theorem, we deduce that the structures $\langle \mathcal{P}(\omega) / \mathrm{fin},\sigma \rangle$ and $\langle \mathcal{P}(\omega) / \mathrm{fin},\sigma^{-1} \rangle$ are elementarily equivalent in the language of algebraic dynamical systems (Boolean algebras together with an automorphism). This corollary does not depend on the Continuum Hypothesis.

math.LO↗
arXiv2023

Cardinal invariants of a meager ideal

Let $\mathcal M_X$ denote the ideal of meager subsets of a topological space $X$. We prove that if $X$ is a completely metrizable space without isolated points, then the smallest cardinality of a non-meager subset of $X$, denoted $\mathrm{non}(\mathcal M_X)$, is exactly $\mathrm{non}(\mathcal M_X) = \mathrm{cf}[\kappa]^\omega \cdot \mathrm{non}(\mathcal M_{\mathbb R})$, where $\kappa$ is the minimum weight of a nonempty open subset of $X$. We also characterize the additivity and covering numbers for $\mathcal M_X$ in terms of simple topological properties of $X$. Some bounds are proved and some questions raised concerning the cofinality of $\mathcal M_X$ and the cofinality of the related ideal of nowhere dense subsets of $X$. We also show that if $X$ is a compact Hausdorff space with $\pi$-weight $\kappa$, then $\mathrm{non}(\mathcal M_X) \leq \mathrm{cf}[\kappa]^\omega \cdot \mathrm{non}(\mathcal M_{\mathbb R})$. This bound for compact Hausdorff spaces is not sharp, in the sense that it is consistent for such a space to have non-meager subsets of even smaller cardinality.

math.GN↗
arXiv2023

On Roitman's principles $\mathsf{MH}$ and $\Delta$

The Model Hypothesis (abbreviated $\mathsf{MH}$) and $\Delta$ are set-theoretic axioms introduced by J. Roitman in her work on the box product problem. Answering some questions of Roitman and Williams on these two principles, we show (1) $\mathsf{MH}$ implies the existence of $P$-points in $\omega^*$ and is therefore not a theorem of $\mathsf{ZFC}$; (2) $\mathsf{MH}$ also fails in the side-by-side Sacks models; (3) as $\Delta$ holds in these models, this implies $\Delta$ is strictly stronger than $\mathsf{MH}$; (4) furthermore, $\Delta$ holds in a large class of forcing extensions in which it was not previously known to hold.

math.GN↗
arXiv2022

Elementary submodels, coding strategies, and an infinite real number game

Matthew Baker investigated, in previous work, an elegant, infinite-length game that may be used to study subsets of real numbers. We present two accessible examples of how an important technique from set theory, or a different technique from infinite game theory, may be used to answer Baker's question on whether this game provides a precise characterization for countable subsets of real numbers, and we connect this game to the well-studied Banach-Mazur game from topology.

math.LO↗
arXiv2022

First-countable Lindel\"of scattered spaces

We study the class of first-countable Lindel\"of scattered spaces, or "FLS" spaces. While every $T_3$ FLS space is homeomorphic to a scattered subspace of $\mathbb Q$, the class of $T_2$ FLS spaces turns out to be surprisingly rich. Our investigation of these spaces reveals close ties to $Q$-sets, Lusin sets, and their relatives, and to the cardinals $\mathfrak{b}$ and $\mathfrak{d}$. Many natural questions about FLS spaces turn out to be independent of $\mathsf{ZFC}$. We prove that there exist uncountable FLS spaces with scattered height $\omega$. On the other hand, an uncountable FLS space with finite scattered height exists if and only if $\mathfrak{b} = \aleph_1$. We prove some independence results concerning the possible cardinalities of FLS spaces, and concerning what ordinals can be the scattered height of an FLS space. Several open problems are included.

math.GN↗
arXiv2021

Partitioning the real line into Borel sets

For which infinite cardinals $\kappa$ is there a partition of the real line $\mathbb R$ into precisely $\kappa$ Borel sets? Hausdorff famously proved that there is a partition of $\mathbb R$ into $\aleph_1$ Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of $\mathbb R$ into Borel sets can be fairly arbitrary. For example, given any $A \subseteq \omega$ with $0,1 \in A$, there is a forcing extension in which $A = \{ n :\, \text{there is a partition of }\mathbb R\text{ into }\aleph_n\text{ Borel sets}\}$. We also look at the corresponding question for partitions of $\mathbb R$ into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable $\kappa$ such that there is a partition of $\mathbb R$ into precisely $\kappa$ closed sets can be fairly arbitrary.

math.LO↗
arXiv2021

Covering versus partitioning with the Cantor space

What topological spaces can be partitioned into copies of the Cantor space $2^\omega$? An obvious necessary condition is that a space can be partitioned into copies of $2^\omega$ only if it can be covered with copies of $2^\omega$. We prove three theorems concerning when this necessary condition is also sufficient. If $X$ is a metrizable space and $|X| \leq \mathfrak{c}^{+\omega}$ (the least limit cardinal $>\!\mathfrak{c}$), then $X$ can be partitioned into copies of $2^\omega$ if and only if $X$ can be covered with copies of $2^\omega$. To show this cardinality bound is sharp, we construct a metrizable space of size $\mathfrak{c}^{+(\omega+1)}$ that can be covered with copies of $2^\omega$, but not partitioned into copies of $2^\omega$. Similarly, if $X$ is first countable and $|X| \leq \mathfrak{c}$, then $X$ can be partitioned into copies of $2^\omega$ if and only if $X$ can be covered with copies of $2^\omega$. On the other hand, there is a first countable space of size $\mathfrak{c}^+$ that can be covered with copies of $2^\omega$, but not partitioned into copies of $2^\omega$. Finally, we show that a completely metrizable space can be partitioned into copies of $2^\omega$ if and only if it can be covered with copies of $2^\omega$ if and only if it has no isolated points.

math.GN↗
arXiv2021

Combinatorial and number-theoretic properties of generic reals

We discuss some properties of Cohen and random reals. We show that they belong to any definable partition regular family, and hence they satisfy most "largeness" properties studied in Ramsey theory. We determine their position in the Mahler's classification of the reals and using it, we get some information about Liouville numbers. We also show that they are wild in the sense of o-minimality, i.e., they define the set of integers.

math.LO↗
arXiv2021

Covering versus partitioning with Polish spaces

Given a completely metrizable space $X$, let $\mathfrak{par}(X)$ denote the smallest possible size of a partition of $X$ into Polish spaces, and $\mathfrak{cov}(X)$ the smallest possible size of a covering of $X$ with Polish spaces. Observe that $\mathfrak{cov}(X) \leq \mathfrak{par}(X)$ for every $X$, because every partition of $X$ is also a covering. We prove it is consistent relative to a huge cardinal that the strict inequality $\mathfrak{cov}(X) < \mathfrak{par}(X)$ can hold for some completely metrizable space $X$. We also prove that using large cardinals is necessary for obtaining this strict inequality, because if $\mathfrak{cov}(X) < \mathfrak{par}(X)$ for any completely metrizable $X$, then $0^\dagger$ exists.

math.LO↗
arXiv2020

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

It was proved recently that Telg\'arsky'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 $\kappa$-cc contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < \kappa$ 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-$\aleph_\omega$ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $\mathsf{ZFC}+\mathsf{GCH}$.

math.LO↗
arXiv2019

Telgarsky's conjecture may fail

Telg\'arsky'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 $\pi$-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 $\pi$-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'' $\pi$-bases mentioned above: \emph{Every separative poset $\mathbb P$ with the $\kappa$-cc contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < \kappa$ 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 $|\mathbb P| < \aleph_\omega$, then \axiom-for-$\mathbb P$ is a consequence of $\mathsf{GCH}$ holding below $|\mathbb P|$.

math.LO↗