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24
Countable dense homogeneity of definable spaces
 PROC. AMER. MATH. SOC
, 2004
"... 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 zerodimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω the following are equivalent: (1) X is Gδ in 2 ..."
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Cited by 10 (1 self)
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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 zerodimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω the following are equivalent: (1) X is Gδ in 2 ω, (2) X ω is CDH and (3) X ω is homeomorphic to 2 ω or to ω ω. 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 large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Steprāns and Zhou by showing that p = min{κ: 2 κ is not CDH}. A separable topological space X is countable dense homogeneous (CDH) if given any two countable dense subsets D, D ′ ⊆ X there is a homeomorphism h of X such that h[D] = D ′. The first result in this area is due to Cantor, who, in effect, showed that the reals are CDH. Fréchet [Fr] and Brower [Br], independently, proved
Cofinal types of topological directed orders
 ANN. INST. FOURIER, GRENOBLE
"... We investigate the structure of the Tukey ordering among directed orders arising naturally in topology and measure theory. ..."
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Cited by 7 (3 self)
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We investigate the structure of the Tukey ordering among directed orders arising naturally in topology and measure theory.
The structure of the σideal of σporous sets
, 1999
"... We show a general method of construction of nonσporous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each nonσporous Suslin subset of a topologically complete metric space contains a nonσporous closed subset. We show also a sufficie ..."
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Cited by 6 (2 self)
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We show a general method of construction of nonσporous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each nonσporous Suslin subset of a topologically complete metric space contains a nonσporous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a nonσporous element. Namely, if we denote the space of all compact subsets of a compact metric space E with the Hausdorff metric by K(E), then it is shown that each analytic subset of K(E) containing all countable compact subsets of E contains necessarily an element, which is nonσporous subset of E. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed nonσporous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the σideal of compact σporous sets.
Descriptive set theory of families of small sets
 J. Symbolic Logic
"... Abstract. This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σideals of closed or compact sets. ..."
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Cited by 6 (0 self)
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Abstract. This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σideals of closed or compact sets.
Trichotomies for ideals of compact sets
 J. SYMBOLIC LOGIC
"... We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal. ..."
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Cited by 5 (3 self)
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We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal.
Settheoretic Problems Concerning Lindelöf Spaces
, 2011
"... I survey problems concerning Lindelöf spaces which have partial settheoretic solutions. Lindelöf spaces, i.e. spaces in which every open cover has a countable subcover, are a familiar class of topological spaces. There is a significant number of (mainly classic) problems concerning Lindelöf spaces w ..."
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Cited by 4 (1 self)
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I survey problems concerning Lindelöf spaces which have partial settheoretic solutions. Lindelöf spaces, i.e. spaces in which every open cover has a countable subcover, are a familiar class of topological spaces. There is a significant number of (mainly classic) problems concerning Lindelöf spaces which are unsolved, but have partial settheoretic solutions. For example, consistency is known but independence is not; large cardinals suffice but are not known to be necessary, and so forth. The purpose of this note — which is an expanded version of a talk given at the 2010 BEST conference — is to survey such questions in the hope that set theorists will find them worthy of attention. Indeed we strongly suspect that the difficulty of these problems is more settheoretic than topological. Not much topological knowledge is needed to work on them. Undefined terms can be found in [Eng89]. In Sections 1–3 we shall assume all spaces are T3, except for a remark at the end of Section 2. I shall start with a collection of problems I have been investigating for the past couple of years. Several of these problems are classic and wellknown; other are more specialized or recent, but are related to the classic ones. 1 Productively Lindelöf spaces Definition 1.1. A space X is productively Lindelöf if X × Y is Lindelöf, for every Lindelöf Y. X is powerfully Lindelöf if X ω is Lindelöf. Both of these concepts have been studied for a long time, but the terminology is recent: [BKR07], [AT] respectively. Ernie Michael wondered more than thirty years ago whether:
Countable dense homogeneity of definable spaces
 Proc. Amer. Math. Soc
"... Abstract. 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 zerodimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω the following are equivalent: (1) X is ..."
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Cited by 3 (0 self)
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Abstract. 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 zerodimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω the following are equivalent: (1) X is Gδ in 2 ω, (2) X ω is CDH and (3) X ω is homeomorphic to 2 ω or to ω ω. 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 large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Steprāns and Zhou by showing that p = min{κ: 2 κ is not CDH}. A separable topological space X is countable dense homogeneous (CDH) if given any two countable dense subsets D, D ′ ⊆ X there is a homeomorphism h of X such that h[D] = D ′. The first result in this area is due to Cantor, who, in effect, showed that the reals are CDH. Fréchet [Fr] and Brower [Br], independently, proved
A Gδ Ideal of Compact Sets Strictly Above the Nowhere Dense
 Ideal in the Tukey Order, Ann. Pure Appl. Logic 156 (2008
"... Abstract. We prove that there is a Gδ σideal of compact sets which is strictly above NWD in the Tukey order. Here NWD is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličkovic ́ asked in [4]. 1. ..."
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Cited by 3 (2 self)
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Abstract. We prove that there is a Gδ σideal of compact sets which is strictly above NWD in the Tukey order. Here NWD is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličkovic ́ asked in [4]. 1.
On the complexity of some σideals of σPporous sets
 COMMENT.MATH.UNIV.CAROLIN. 44,3 (2003)531–554 531
, 2003
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