arXiv subjects

David Milovich

Publications and source records attributed to David Milovich.

15 recordsLinked to original sources

arXiv2019

Between homeomorphism type and Tukey type

Call a compact space $X$ pin homogeneous if every two points $a,b$ are pin equivalent, meaning that there exists a compact space $Y$, a quotient map $f\colon Y\to X$, and a homeomorphism $g\colon Y\to Y$ such that $gf^{-1}\{a\}=f^{-1}\{b\}$. We will prove a representation theorem for pin equivalence; transitivity of pin equivalence will be a corollary. Pin homogeneity is strictly weaker than homogeneity and pin equivalence is strictly stronger than Tukey equivalence. Just as with topological homogeneity, no infinite compact $F$-space is pin homogeneous. On the other hand, $X\times 2^{χ(X)}$ is pin homogeneous for every compact $X$. And there is a compact pin homogeneous space with points of different $π$-character.

math.GN↗

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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arXiv2017

Non-Absoluteness of Model Existence at $\aleph_ω$

In [FHK13], the authors considered the question whether model-existence of $L_{ω_1,ω}$-sentences is absolute for transitive models of ZFC, in the sense that if $V \subseteq W$ are transitive models of ZFC with the same ordinals, $φ\in V$ and $V\models "φ\text{ is an } L_{ω_1,ω}\text{-sentence}"$, then $V \models "φ\text{ has a model of size } \aleph_α"$ if and only if $W \models "φ\text{ has a model of size } \aleph_α"$. From [FHK13] we know that the answer is positive for $α=0,1$ and under the negation of CH, the answer is negative for all $α>1$. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each $α>1$, except the case when $α=ω$ which is an open question in [FHK13]. We answer the open question by providing a negative answer under GCH even for $α=ω$. Our examples are incomplete sentences. In fact, the same sentences can be used to prove a negative answer under GCH for all $α>1$ assuming the consistency of a Mahlo cardinal. Thus, the large cardinal assumption is relaxed from a supercompact in [FHK13] to a Mahlo cardinal. Finally, we consider the absoluteness question for the $\aleph_α$-amalgamation property of $L_{ω_1,ω}$-sentences (under substructure). We prove that assuming GCH, $\aleph_α$-amalgamation is non-absolute for $1<α<ω$. This answers a question from [SS]. The cases $α=1$ and $α$ infinite remain open. As a corollary we get that it is non-absolute that the amalgamation spectrum of an $L_{ω_1,ω}$-sentence is empty.

math.LO↗

arXiv2016

Chaos and periodicity on star graphs

For a continuous self-map of a star graph to be Li-Yorke chaotic and to have full periodicity, we prove some new sufficient conditions on the orbit of the center.

math.DS↗

arXiv2016

On the strong Freese-Nation property

We show that there is a boolean algebra that has the Freese-Nation property (FN) but not the strong Freese-Nation property (SFN), thus answering a question of Heindorf and Shapiro. Along the way, we produce some new characterizations of the FN and SFN in terms of sequences of elementary submodels.

math.LO↗

arXiv2016

Team games, hypergraph spaces, and projective Boolean algebras

We modify the game Fuchino, Koppelberg, and Shelah used to characterize the $\kappa$-Freese-Nation property for a given Boolean algebra $A$, replacing players I and II each with a team of $n$ players with limited information. We show that $A$ is tightly $\kappa$-filtered exactly when team II has a winning strategy for every finite team size. Case $\kappa=\aleph_0$ characterizes projective Boolean algebras and, hence, Dugundji spaces. In terms of the open-open game of Daniels, Kunen, and Zhou, this characterization is a team version of very I-favorable. We similarly characterize Cohen algebras in terms of a team version of I-favorability. If $A$ is the clopen algebra of the space of $n$-uniform hypergraphs on $\kappa^{+n}$ that avoid copies of $[n+1]^n$, then team II has a winning strategy for our modified FKS game for team size $n-1$ but not $n$. For $n\geq 3$, this algebra also answers a question of Geschke when combined with a locally ${<}\kappa$-sized characterization of tightly $\kappa$-filtered Boolean algebras that we prove. Case $\kappa=\aleph_0$ includes a locally finite characterization of projective Boolean algebras.

math.LO↗

arXiv2012

Forbidden rectangles in compacta

We establish negative results about "rectangular" local bases in compacta. For example, there is no compactum where all points have local bases of cofinal type ωx ω_2. For another, the compactum βωhas no nontrivially rectangular local bases, and the same is consistently true of βω\ ω: no local base in βωhas cofinal type κx c if κ< m_{σ-n-linked} for some n in [1,ω). Also, CH implies that every local base in βω\ ωhas the same cofinal type as one in βω. We also answer a question of Dobrinen and Todorcevic about cofinal types of ultrafilters: the Fubini square of a filter on ωalways has the same cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal P-filters on ωis commutative modulo cofinal equivalence.

math.GN↗

arXiv2012

Noetherian type in topological products

The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_ω$ with the countable box topology, $(2^{\aleph_ω})_δ$, is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of $\aleph_ω$. We discuss the influence of principles like $\square_{\aleph_ω}$ and Chang's conjecture for $\aleph_ω$ on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an $(\aleph_4,\aleph_1)$-sparse covering family of countable subsets of $\aleph_ω$. From this follows an absolute upper bound of $\aleph_4$ on the Noetherian type of $(2^{\aleph_ω})_δ$. The proof uses ideas from Shelah's proof that if $κ^+ <λ$ then his ideal $I[λ]$ contains a stationary set consisting of points of cofinality $κ$.

math.GN↗

arXiv2012

The (λ, κ)-Freese-Nation property for boolean algebras and compacta

We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite κ, the (κ,κ)-FN, the (κ^+,κ)-FN, and the κ-FN are known to be equivalent; we show that the family of properties (λ,μ)-FN for λ>μform a true two-dimensional hierarchy that is robust with respect to coproducts, retracts, and the exponential operation. The (κ,\aleph_0)-FN in particular has strong consequences for base properties of compacta (stronger still for homogeneous compacta), and these consequences have natural duals in terms of special subsets of boolean algebras. We show that the (κ,\aleph_0)-FN also leads to a generalization of the equality of weight and π-character in dyadic compacta. Elementary subalgebras and their duals, elementary quotient spaces, were originally used to define the (λ, κ)-FN and its topological dual, which naturally generalized from Stone spaces to all compacta, thereby generalizing Shchepin's notion of openly generated compacta. We introduce a simple combinatorial definition of the (λ, κ)-FN that is equivalent to the original for regular infinite cardinals λ>κ.

math.LO↗

arXiv2011

The topology of ultrafilters as subspaces of $2^ω$

Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin's Axiom for countable posets, to distinguish non-principal ultrafilters on $ω$ up to homeomorphism. Here, we identify ultrafilters with subpaces of $2^ω$ in the obvious way. Using the same methods, still under Martin's Axiom for countable posets, we will construct a non-principal ultrafilter $\UU\subseteq 2^ω$ such that $\UU^ω$ is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial results about the relation of such topological properties with the combinatorial property of being a $\mathrm{P}$-point.

math.GN↗

arXiv2010

GO-spaces and Noetherian spectra

The Noetherian type of a space is the least k for which the space has a k^op-like base, i.e., a base in which no element has k-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products thereof. For example: the density of a product of not-too-many compact linear orders never exceeds its Noetherian type, with equality possible only for singular Noetherian types; we prove a similar result for products of Lindelof GO-spaces. A countable product of compact linear orders has an omega_1^op-like base if and only if it is metrizable, and every metrizable space has an omega^op-like base. An infinite cardinal k is the Noetherian type of a compact LOTS if and only if k is not omega_1 and k is not weakly inaccessible. There is a Lindelof LOTS with Noetherian type omega_1 and there consistently is a Lindelof LOTS with weakly inaccessible Noetherian type.

math.GN↗

arXiv2008

Tukey classes of ultrafilters on omega

Motivated by a question of Isbell, we show that Jensen's Diamond Principle implies there is a non-P-point ultrafilter U on omega such that U, whether ordered by reverse inclusion or reverse inclusion mod finite, is not Tukey equivalent to the finite sets of reals ordered by inclusion. We also show that, for every regular infinite kappa not greater than 2^{aleph_0}, if MA_{sigma-centered} holds, then some ultrafilter U on omega, ordered by reverse inclusion mod finite, is Tukey equivalent to the sets of reals of size less than kappa, ordered by inclusion. We also prove two negative ZFC results about the possible Tukey classes of ultrafilters on omega.

math.LO↗

arXiv2008

Splitting families and the Noetherian type of $βω-ω$

Extending some results of Malykhin, we prove several independence results about base properties of $βω-ω$ and its powers, especially the Noetherian type $Nt(βω-ω)$, the least $κ$ for which $βω-ω$ has a base that is $κ$-like with respect to containment. For example, $Nt(βω-ω)$ is never less than the splitting number, but can consistently be that $ω_1$, $2^ω$, $(2^ω)^+$, or strictly between $ω_1$ and $2^ω$. $Nt(βω-ω)$ is also consistently less than the additivity of the meager ideal. $Nt(βω-ω)$ is closely related to the existence of special kinds of splitting families.

math.LO↗

arXiv2007

Noetherian types of homogeneous compacta and dyadic compacta

The Noetherian type of a space is the least $κ$ such that it has a base that is $κ$-like with respect to containment. Just as all known homogeneous compacta have cellularity at most $2^ω$, they satisfy similar upper bounds in terms of Noetherian type and related cardinal functions. We prove these and many other results about these cardinal functions. For example, every homogeneous dyadic compactum has Noetherian type $ω$. Assuming GCH, every point in a homogeneous compactum $X$ has a local base that is $c(X)$-like with respect to containment. If every point in a compactum has a well-quasiordered local base, then some point has a countable local $π$-base.

math.GN↗

arXiv2006

Amalgams, connectifications, and homogeneous compacta

We construct a path-connected homogenous compactum with cellularity 2^omega that is not homeomorphic to any product of dyadic compacta and first countable compacta. We also prove some closure properties for classes of spaces defined by various connectifiability conditions. One application is that every infinite product of infinite topological sums of T_i spaces has a T_i pathwise connectification, where i is 1, 2, 3, or 3.5.

math.GN↗