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Papers by michael hrusak

Journal of Mathematical Logic

A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong ho... more A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits [Formula: see text] [Formula: see text] of a certain inverse system [Formula: see text] indexed by [Formula: see text] to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all [Formula: see text]-coherent families of functions indexed by [Formula: see text] to be trivial. In this paper, we prove that, in any forcing extension given by adjoining [Formula: see text]-many Cohen reals, [Formula: see text] vanishes for all [Formula: see text]. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional [Formula: see text]-system lemmas. Thi...

Transactions of the American Mathematical Society, 2003

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

Proceedings of the American Mathematical Society, 2005

We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We ... more We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω X\subseteq 2^{\omega } the following are equivalent: (1) X X is G δ G_{\delta } in 2 ω 2^{\omega } , (2) X ω X^{\omega } is CDH and (3) X ω X^{\omega } is homeomorphic to 2 ω 2^{\omega } or to ω ω \omega ^{\omega } . Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr a ¯ \bar {\text {a}} ns and Zhou, by showing that p = min { κ : 2 κ \mathfrak {p}= \min \{\kappa : 2^{\kappa } is not CDH } \} .

Applied General Topology, 2020

In this short note we prove the existence (in ZFC) of a completely regular countable disjointly t... more In this short note we prove the existence (in ZFC) of a completely regular countable disjointly tight irresolvable space by showing that every sub-maximal countable dense subset of 2c is disjointly tight.

Topology and its Applications, 2005

We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We... more We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We 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 show that Mathias--Prikry forcing associated with the complement of an analytic ideal does add a dominating real. We give a characterization of omega\omegaomega-hitting and omega\omegaomega-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.

Answering a question of Marion Scheepers, we show that the cardinal invariant 𝔡 is a lower bound ... more Answering a question of Marion Scheepers, we show that the cardinal invariant 𝔡 is a lower bound on 𝔦rr, the minimal weight (equivalently, minimal π-weight) of a countable regular irresolvable space. We consider related cardinal invariants such as r scat , the reaping number of the quotient algebra P(ℚ) mod the ideal of scattered subsets of the rationals, and prove that ⋄(r scat ) implies that 𝔦rr=ω 1 .

Topology and its Applications, 2012

Topology and its Applications, 2007

Topology and its Applications, 2005

Topology and its Applications, 2005

Journal of Symbolic Logic, 2003

An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of ... more An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size c and decreasing chains of length c+ bellow every element. Assuming t = c a MAD family equivalent to all of its restrictions is constructed. It is also shown here that the Continuum Hypothesis implies that for every ωω-bounding forcing ℙ of size c there is a Cohen-destructible, ℙ-indestructible MAD family. Finally, two other orderings on MAD families are suggested and an old construction of Mrówka is revisited.

The Journal of Symbolic Logic, 2012

A metric space (X, d) ismonotoneif there is a linear order < onXand a constantcsuch thatd(x, y... more A metric space (X, d) ismonotoneif there is a linear order < onXand a constantcsuch thatd(x, y)≤c d(x, z)for allx

The Journal of Symbolic Logic, 2010

We investigate the pair-splitting numberwhich is a variation of splitting number, pair-reaping nu... more We investigate the pair-splitting numberwhich is a variation of splitting number, pair-reaping numberwhich is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants ofFσ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

The Journal of Symbolic Logic, 2009

It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable.... more It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].

Journal of Symbolic Logic, 2001

In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofini... more In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(ω) is strictly greater than the cardinal numbers and .

Fundamenta Mathematicae, 2009

Fundamenta Mathematicae, 2004

Journal of Mathematical Logic

A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong ho... more A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits [Formula: see text] [Formula: see text] of a certain inverse system [Formula: see text] indexed by [Formula: see text] to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all [Formula: see text]-coherent families of functions indexed by [Formula: see text] to be trivial. In this paper, we prove that, in any forcing extension given by adjoining [Formula: see text]-many Cohen reals, [Formula: see text] vanishes for all [Formula: see text]. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional [Formula: see text]-system lemmas. Thi...

Transactions of the American Mathematical Society, 2003

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

Proceedings of the American Mathematical Society, 2005

We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We ... more We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω X\subseteq 2^{\omega } the following are equivalent: (1) X X is G δ G_{\delta } in 2 ω 2^{\omega } , (2) X ω X^{\omega } is CDH and (3) X ω X^{\omega } is homeomorphic to 2 ω 2^{\omega } or to ω ω \omega ^{\omega } . Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr a ¯ \bar {\text {a}} ns and Zhou, by showing that p = min { κ : 2 κ \mathfrak {p}= \min \{\kappa : 2^{\kappa } is not CDH } \} .

Applied General Topology, 2020

In this short note we prove the existence (in ZFC) of a completely regular countable disjointly t... more In this short note we prove the existence (in ZFC) of a completely regular countable disjointly tight irresolvable space by showing that every sub-maximal countable dense subset of 2c is disjointly tight.

Topology and its Applications, 2005

We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We... more We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We 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 show that Mathias--Prikry forcing associated with the complement of an analytic ideal does add a dominating real. We give a characterization of omega\omegaomega-hitting and omega\omegaomega-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.

Answering a question of Marion Scheepers, we show that the cardinal invariant 𝔡 is a lower bound ... more Answering a question of Marion Scheepers, we show that the cardinal invariant 𝔡 is a lower bound on 𝔦rr, the minimal weight (equivalently, minimal π-weight) of a countable regular irresolvable space. We consider related cardinal invariants such as r scat , the reaping number of the quotient algebra P(ℚ) mod the ideal of scattered subsets of the rationals, and prove that ⋄(r scat ) implies that 𝔦rr=ω 1 .

Topology and its Applications, 2012

Topology and its Applications, 2007

Topology and its Applications, 2005

Topology and its Applications, 2005

Journal of Symbolic Logic, 2003

An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of ... more An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size c and decreasing chains of length c+ bellow every element. Assuming t = c a MAD family equivalent to all of its restrictions is constructed. It is also shown here that the Continuum Hypothesis implies that for every ωω-bounding forcing ℙ of size c there is a Cohen-destructible, ℙ-indestructible MAD family. Finally, two other orderings on MAD families are suggested and an old construction of Mrówka is revisited.

The Journal of Symbolic Logic, 2012

A metric space (X, d) ismonotoneif there is a linear order < onXand a constantcsuch thatd(x, y... more A metric space (X, d) ismonotoneif there is a linear order < onXand a constantcsuch thatd(x, y)≤c d(x, z)for allx

The Journal of Symbolic Logic, 2010

We investigate the pair-splitting numberwhich is a variation of splitting number, pair-reaping nu... more We investigate the pair-splitting numberwhich is a variation of splitting number, pair-reaping numberwhich is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants ofFσ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

The Journal of Symbolic Logic, 2009

It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable.... more It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].

Journal of Symbolic Logic, 2001

In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofini... more In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(ω) is strictly greater than the cardinal numbers and .

Fundamenta Mathematicae, 2009

Fundamenta Mathematicae, 2004