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The Steel Hierarchy of Ordinal Valued Borel Mappings
, 2003
"... Given well ordered countable sets of the form , we consider Borel mappings from ! with countable image inside the ordinals. The ordinals and ! are respectively equipped with the discrete topology and the product of the discrete topology on . The Steel wellordering on such mappings is de ..."
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Given well ordered countable sets of the form , we consider Borel mappings from ! with countable image inside the ordinals. The ordinals and ! are respectively equipped with the discrete topology and the product of the discrete topology on . The Steel wellordering on such mappings is dened by i there exists a continuous function f such that (x) (x) holds for any x 2 ! . It induces a hierachy of mappings which we give a complete description of. We provide, for each ordinal , a mapping T() whose rank is precisely in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by . These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy.
Descriptive Kakutani equivalence
, 2005
"... Abstract. We consider a descriptive version of Kakutani equivalence for Borel automorphisms of Polish spaces. Answering a question of Nadkarni, we show that up to this notion, there are exactly two Borel automorphisms: those which are smooth, and those which are not. Using this, we classify all Bore ..."
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Abstract. We consider a descriptive version of Kakutani equivalence for Borel automorphisms of Polish spaces. Answering a question of Nadkarni, we show that up to this notion, there are exactly two Borel automorphisms: those which are smooth, and those which are not. Using this, we classify all Borel Rflows up to C ∞ timechange isomorphism. We then extend the notion of Kakutani equivalence to all (not necessarily injective) Borel functions, and provide a variety of results leading towards a full classification of Borel functions on Polish spaces. The main technical tools are a series of GlimmEffros and DoughertyJacksonKechris style embedding theorems. 1.
BOREL STRUCTURES: A BRIEF SURVEY
"... Abstract. We survey some research aiming at a theory of effective structures of size the continuum. The main notion is the one of a Borel presentation, where the domain, equality and further relations and functions are Borel. We include the case of uncountable languages where the signature is Borel. ..."
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Abstract. We survey some research aiming at a theory of effective structures of size the continuum. The main notion is the one of a Borel presentation, where the domain, equality and further relations and functions are Borel. We include the case of uncountable languages where the signature is Borel. We discuss the main open questions in the area. 1.
ON ANTICHAINS OF SPREADING MODELS OF BANACH SPACES
, 805
"... Abstract. We show that for every separable Banach space X, either SPw(X) (the set of all spreading models of X generated by weaklynull sequences in X, modulo equivalence) is countable, or SPw(X) contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell ..."
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Abstract. We show that for every separable Banach space X, either SPw(X) (the set of all spreading models of X generated by weaklynull sequences in X, modulo equivalence) is countable, or SPw(X) contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell and B. Sari. 1.
S07644442(00)016207/FLA AID:1620 Vol.331(0) P.1(16) CRAcad 2000/06/15 Prn:1/08/2000; 14:04 F:PXMA1620.tex by:EL p. 1
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The Steel Hierarchy of Ordinal Valued Borel Mappings
"... 1 Introduction We deal with mappings from!sequences of elements from countable wellordered alphabets, into the ordinals. Given the discrete topology on ordinals and the usual topology on sets of the form \Lambda! (the product topology of the discrete topology on \Lambda), we are only concerned wit ..."
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1 Introduction We deal with mappings from!sequences of elements from countable wellordered alphabets, into the ordinals. Given the discrete topology on ordinals and the usual topology on sets of the form \Lambda! (the product topology of the discrete topology on \Lambda), we are only concerned with Borel mappings with countable image for reasons of using determinacy without large cardinal hypothesis. The Steel ordering between two such mappings OE: \Lambda OE!
The following proposition answers the question of the relation between cardinalities of antichains of P and antichains of F? Proposition 1
"... 1. Every antichain in P has cardinality 1 =) every antichain in F has cardinality 1 2. There exists an antichain in P of cardinality 2, but no element of P is incomparable with two different elements =) every antichain in F has cardinality at most 2 3. There exists an element in P which is incom ..."
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1. Every antichain in P has cardinality 1 =) every antichain in F has cardinality 1 2. There exists an antichain in P of cardinality 2, but no element of P is incomparable with two different elements =) every antichain in F has cardinality at most 2 3. There exists an element in P which is incomparable with two different elements =) there exists antichains of any cardinality in F.