scholarly journals CONTINUOUS RAMSEY THEORY ON POLISH SPACES AND COVERING THE PLANE BY FUNCTIONS

2004 ◽  
Vol 04 (02) ◽  
pp. 109-145 ◽  
Author(s):  
STEFAN GESCHKE ◽  
MARTIN GOLDSTERN ◽  
MENACHEM KOJMAN
Keyword(s):  
Set Theory ◽  
Metric Spaces ◽  
Ramsey Theory ◽  
Polish Space ◽  
Lipschitz Functions ◽  
Cantor Space ◽  
Polish Spaces ◽  
Real Functions ◽  
Number Formula ◽  
Independence Results

We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: see text] of a pair-coloring c:[X]2→2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2ω, c min and c max , which satisfy [Formula: see text] and prove: Theorem. (1) For every Polish space X and every continuous pair-coloringc:[X]2→2with[Formula: see text], [Formula: see text] (2) There is a model of set theory in which[Formula: see text]and[Formula: see text]. The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. We prove that [Formula: see text] is equal to the covering number of (2ω)2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c min gives: Theorem. There is a model of set theory in which (1) ℝ2 is coverable byℵ1graphs and reflections of graphs of continuous real functions; (2) ℝ2 is not coverable byℵ1graphs and reflections of graphs of Lipschitz real functions. Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row (3).

Classifying Borel automorphisms

2007 ◽  
Vol 72 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
John D. Clemens
Keyword(s):  
Polish Space ◽  
Inverse Image ◽  
Product Topology ◽  
Cantor Space ◽  
Space Forms ◽  
Borel Set ◽  
Polish Spaces ◽  
Countable Space ◽  
Fixed Space ◽  
Borel Automorphism

§1. Introduction. This paper considers several complexity questions regarding Borel automorphisms of a Polish space. Recall that a Borel automorphism is a bijection of the space with itself whose graph is a Borel set (equivalently, the inverse image of any Borel set is Borel). Since the inverse of a Borel automorphism is another Borel automorphism, as is the composition of two Borel automorphisms, the set of Borel automorphisms of a given Polish space forms a group under the operation of composition. We can also consider the class of automorphisms of all Polish spaces. We will be primarily concerned here with the following notion of equivalence:Definition 1.1. Two Borel automorphisms f and g of the Polish spaces X and Y are said to be Borel isomorphic, f ≅ g, if they are conjugate, i.e. there is a Borel bijection φ: X → Y such that φ ∘ f = g ∘ φ.We restrict ourselves to automorphisms of uncountable Polish spaces, as the Borel automorphisms of a countable space are simply the permutations of the space. Since any two uncountable Polish spaces are Borel isomorphic, any Borel automorphism is Borel isomorphic to some automorphism of a fixed space. Hence, up to Borel isomorphism we can fix a Polish space and represent any Borel automorphism as an automorphism of this space. We will use the Cantor space 2ω (with the product topology) as our representative space.

scholarly journals A comparison of concepts from computable analysis and effective descriptive set theory

2016 ◽  
Vol 27 (8) ◽  
pp. 1414-1436 ◽  
Author(s):  
VASSILIOS GREGORIADES ◽  
TAMÁS KISPÉTER ◽  
ARNO PAULY
Keyword(s):  
Set Theory ◽  
Metric Spaces ◽  
Computable Analysis ◽  
Descriptive Set Theory ◽  
Polish Spaces ◽  
Complete Metric Spaces ◽  
Cauchy Completion ◽  
Precise Relationship ◽  
Descriptive Set ◽  
Effective Descriptive Set Theory

Computable analysis and effective descriptive set theory are both concerned with complete metric spaces, functions between them and subsets thereof in an effective setting. The precise relationship of the various definitions used in the two disciplines has so far been neglected, a situation this paper is meant to remedy.As the role of the Cauchy completion is relevant for both effective approaches to Polish spaces, we consider the interplay of effectivity and completion in some more detail.

scholarly journals Turing degrees in Polish spaces and decomposability of Borel functions

2020 ◽  
Vol 21 (01) ◽  
pp. 2050021
Author(s):  
Vassilios Gregoriades ◽  
Takayuki Kihara ◽  
Keng Meng Ng
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Recursion Theory ◽  
Descriptive Set Theory ◽  
Analytic Subset ◽  
Polish Spaces ◽  
Negative Results ◽  
Borel Functions ◽  
Important Open Problem ◽  
Descriptive Set

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture.

scholarly journals GREY SUBSETS OF POLISH SPACES

2015 ◽  
Vol 80 (4) ◽  
pp. 1379-1397 ◽  
Author(s):  
ITAÏ BEN YAACOV ◽  
JULIEN MELLERAY
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Descriptive Set Theory ◽  
Polish Spaces ◽  
Small Index Property ◽  
Small Index ◽  
Descriptive Set

AbstractWe develop the basics of an analogue of descriptive set theory for functions on a Polish space X. We use this to define a version of the small index property in the context of Polish topometric groups, and show that Polish topometric groups with ample generics have this property. We also extend classical theorems of Effros and Hausdorff to the topometric context.

Ergodic decompositions associated with regular Markov operators on Polish spaces

2010 ◽  
Vol 31 (2) ◽  
pp. 571-597 ◽  
Author(s):  
DANIËL T. H. WORM ◽  
SANDER C. HILLE
Keyword(s):  
State Space ◽  
Metric Spaces ◽  
Transition Probabilities ◽  
Polish Space ◽  
Invariant Probability Measure ◽  
The State ◽  
Probability Measures ◽  
Ergodic Decomposition ◽  
Polish Spaces ◽  
Appl Math

AbstractFor any regular Markov operator on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrization of the ergodic probability measures associated with this operator in terms of particular subsets of the state space. We use this parametrization to prove an integral decomposition of every invariant probability measure in terms of the ergodic probability measures and give an ergodic decomposition of the state space. This extends results by Yosida [Functional Analysis. Springer, Berlin, 1980, Ch. XIII.4], Hernández-Lerma and Lasserre [Ergodic theorems and ergodic decomposition for Markov chains. Acta Appl. Math.54 (1998), 99–119] and Zaharopol [An ergodic decomposition defined by transition probabilities. Acta Appl. Math.104 (2008), 47–81], who considered the setting of locally compact separable metric spaces. Our extension to Polish spaces solves an open problem posed by Zaharopol (loc. cit.) in a satisfactory manner.

Topological Manifolds and Polish Spaces

2021 ◽  
Author(s):  
Yu-Lin Chou
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Open Cover ◽  
Descriptive Set Theory ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Polish Spaces ◽  
Manifold With Boundary ◽  
Topological Manifolds ◽  
Descriptive Set

We give,as a preliminary result, some topological characterizations of locally compact second-countable Hausdorff spaces. Then we show that a topological manifold, with boundary or not,is precisely a Polish space with a coordinate open cover; this connects geometry with descriptive set theory.

scholarly journals Some Random Coupled Best Proximity Points for a Generalized ω-Cyclic Contraction in Polish Spaces

2017 ◽  
Vol 59 (1) ◽  
pp. 91-105 ◽  
Author(s):  
C. Kongban ◽  
P. Kumam
Keyword(s):  
Metric Spaces ◽  
Best Proximity Point ◽  
Cyclic Contraction ◽  
Polish Spaces ◽  
Best Proximity Points ◽  
Coupled Best Proximity Point

AbstractIn this paper, we will introduce the concepts of a random coupled best proximity point and then we prove the existence of random coupled best proximity points in separable metric spaces. Our results extend the previous work of Akbar et al.[1].

scholarly journals Two classes of metric spaces

2016 ◽  
Vol 17 (1) ◽  
pp. 57 ◽  
Author(s):  
Isabel Garrido ◽  
Ana S. Meroño
Keyword(s):  
Metric Spaces ◽  
Lipschitz Functions ◽  
The Other ◽  
Metric Properties ◽  
Other Hand ◽  
Common Framework

<p>The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.</p>

PRIMITIVE INDEPENDENCE RESULTS

2003 ◽  
Vol 03 (01) ◽  
pp. 67-83
Author(s):  
HARVEY M. FRIEDMAN
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Power Set ◽  
Independence Results

We present some new set and class theoretic independence results from ZFC and NBGC that are particularly simple and close to the primitives of membership and equality (see Secs. 4 and 5). They are shown to be equivalent to familiar small large cardinal hypotheses. We modify these independendent statements in order to give an example of a sentence in set theory with 5 quantifiers which is independent of ZFC (see Sec. 6). It is known that all 3 quantifier sentences are decided in a weak fragment of ZF without power set (see [4]).

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