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Michael Hrušák

Publications and source records attributed to Michael Hrušák.

28 records · Page 2Linked to original sources
arXiv2018

Spaces of Remote Points

Given a Tychonoff space $X$, let $\varrho(X)$ be the set of remote points of $X$. We view $\varrho(X)$ as a topological space. In this paper we assume that $X$ is metrizable and ask for conditions on $Y$ so that $\varrho(X)$ is homeomorphic to $\varrho(Y)$. This question has been studied before by R. G. Woods and C. Gates. We give some results of the following type: if $X$ has topological property $\mathbf{P}$ and $\varrho(X)$ is homeomorphic to $\varrho(Y)$, then $Y$ also has $\mathbf{P}$. We also characterize the remote points of the rationals and irrationals up to some restrictions. Further, we show that $\varrho(X)$ and $\varrho(Y)$ have open dense homeomorphic subspaces if $X$ and $Y$ are both nowhere locally compact, completely metrizable and share the same cellular type, a cardinal invariant we define.

math.GN↗
arXiv2018

Countable dense homogeneity and $λ$-sets

We show that all sufficiently nice $λ$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $κ\le \mathfrak{b}$ there is a countable dense homogeneous metric space of size $κ$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $κ$ is equivalent to the existence of a $λ$-set of size $κ$. On the other hand, it is consistent with the continuum arbitrarily large that every $\mathsf{CDH}$ metric space has size either $ω_1$ or size $\mathfrak c$. An example of a Baire $\mathsf{CDH}$ metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable $\mathsf{CDH}$ space in ZFC.

math.GN↗
arXiv2017

A parametrized diamond principle and union ultrafilters

We consider a cardinal invariant closely related to Hindman's theorem. We prove that this cardinal invariant is small in the iterated Sacks perfect set forcing model, and that its corresponding parametrized diamond principle implies the existence of union ultrafilters. As a corollary, this establishes the existence of union ultrafilters in the iterated Sacks model of Set Theory.

math.LO↗
arXiv2017

Gruff Ultrafilters

We investigate the question of whether $\mathbb Q$ carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen). We prove that one can (consistently) obtain an affirmative answer to this question in three different ways: by assuming a certain parametrized diamond principle, from the cardinal invariant equality $\mathfrak d=\mathfrak c$, and in the Random real model. [Edit: replace "the Randor real model" with "the model obtained by adding $ω_1$ Cohen reals to a model of $\mathsf{CH}$, and subsequently forcing with the Random algebra"; this is clarified in the corrigendum attached at the end of the paper.]

math.LO↗
arXiv2016

Mathias--Prikry and Laver type forcing; Summable ideals, coideals, and $+$-selective filters

We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias--Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias--Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of $ω$-hitting and $ω$-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.

math.LO↗
arXiv2013

Base Tree Property

Building on previous work of [BPS] we investigate $σ$-closed partial orders of size continuum. We provide both an internal and external characterization of such partial orders by showing that (1) every $σ$-closed partial order of size continuum has a base tree and that (2) $σ$-closed forcing notions of density $\mathfrak c$ correspond exactly to regular suborders of the collapsing algebra $Coll(ω_1, 2^ω)$. We further study some naturally ocurring examples of such partial orders.

math.LO↗
arXiv2012

Properties of functions with monotone graphs

A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y) < c d(x,z) for all x<y<z in X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It is shown, e.g., that such a function can be almost nowhere differentiable, but must be differentiable at a dense set, and that Hausdorff dimension of the graph of such a function is 1.

math.CA↗
arXiv2006

Parametrized $\diamondsuit$ principles

We will present a collection of guessing principles which have a similar relationship to $\diamond$ as cardinal invariants of the continuum have to $\CH$. The purpose is to provide a means for systematically analyzing $\diamond$ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of $\CH$ and $\diamond$ in models such as those of Laver, Miller, and Sacks.

math.LO↗