Louveau's theorem for the descriptive set theory of internal sets

1997 ◽  
Vol 62 (2) ◽  
pp. 595-607
Author(s):  
Kenneth Schilling ◽  
Boško Živaljević
Keyword(s):  
Polish Space ◽  
Borel Subset ◽  
Measurable Space ◽  
Borel Sets ◽  
Polish Spaces ◽  
Borel Hierarchy ◽  
Open Questions ◽  
Vertical Sections ◽  
Almost All ◽  
Internal Sets

AbstractWe give positive answers to two open questions from [15]. (1) For every set C countably determined over , if C is then it must be over , and (2) every Borel subset of the product of two internal sets X and Y all of whose vertical sections are can be represented as an intersection (union) of Borel sets with vertical sections of lower Borel rank. We in fact show that (2) is a consequence of the analogous result in the case when X is a measurable space and Y a complete separable metric space (Polish space) which was proved by A. Louveau and that (1) is equivalent to the property shared by the inverse standard part map in Polish spaces of preserving almost all levels of the Borel hierarchy.

scholarly journals Omega-Polish spaces and measurable selections

1977 ◽  
Vol 23 (3) ◽  
pp. 257-265 ◽  
Author(s):  
Le Van Tu
Keyword(s):  
Large Class ◽  
Banach Spaces ◽  
Polish Space ◽  
Measurable Space ◽  
Selection Theorem ◽  
Polish Spaces ◽  
Measurable Selections

AbstractIn this paper, the author introduces the notion of Ω-Polish spaces (which includes the Polish spaces and a large class of Banach spaces) and extends Castaing's selection theorem (1966) for closed-valued measurable thin multifunctions from a measurable space into an Ω-Polish space. He also extends Robertson's theorem (1974) in the same way.

Boolean operations, Borel sets, and Hausdorff's question

1996 ◽  
Vol 61 (4) ◽  
pp. 1287-1304
Author(s):  
Abhijit Dasgupta
Keyword(s):  
Polish Space ◽  
Descriptive Set Theory ◽  
Boolean Operation ◽  
Boolean Operations ◽  
Borel Sets ◽  
Systematic Fashion ◽  
Borel Hierarchy ◽  
The Difference ◽  
Projective Hierarchy ◽  
Descriptive Set

The study of infinitary Boolean operations was undertaken by the early researchers of descriptive set theory soon after Suslin's discovery of the important operation. The first attempt to lay down their theory in a systematic fashion was the work of Kantorovich and Livenson [5], where they call these the analytical operations. Earlier, Hausdorff had introduced the δs operations — essentially same as the monotoneω-ary Boolean operations, and Kolmogorov, independently of Hausdorff, had discovered the same objects, which were used in his study of the R operator.The ω-ary Boolean operations turned out to be closely related to most of the classical hierarchies over a fixed Polish space X, including, e. g., the Borel hierarchy (), the difference hierarchies of Hausdorff (Dη()), the C-hierarchy (Cξ) of Selivanovski, and the projective hierarchy (): for each of these hierarchies, every level can be expressed as the range of an ω-ary Boolean operation applied to all possible sequences of open subsets of X. In the terminology of Dougherty [3], every level is “open-ω-Boolean” (if and are collections of subsets of X and I is any set, is said to be -I-Boolean if there exists an I-ary Boolean operation Φ such that = Φ, i. e. is the range of Φ restricted to all possible I-sequences of sets from ). If in addition, the space X has a basis consisting of clopen sets, then the levels of the above hierarchies are also “clopen-ω-Boolean.”

scholarly journals Fubini Property for Microscopic Sets

2016 ◽  
Vol 65 (1) ◽  
pp. 143-149
Author(s):  
Adam Paszkiewicz ◽  
Elżbieta Wagner-Bojakowska
Keyword(s):  
Borel Subset ◽  
Horizontal Section ◽  
Polish Spaces ◽  
Vertical Sections

Abstract In 2000, I. Recław and P. Zakrzewski introduced the notion of Fubini Property for the pair (I,J) of two σ-ideals in the following way. Let I and J be two σ-ideals on Polish spaces X and Y, respectively. The pair (I,J) has the Fubini Property (FP) if for every Borel subset B of X×Y such that all its vertical sections Bx = {y ∈ Y : (x, y) ∈ B} are in J, then the set of all y ∈ Y, for which horizontal section By = {x ∈ X : (x, y) ∈ B} does not belong to I, is a set from J, i.e., {y ∈ Y : By ∉ I} ∈ J. The Fubini property for the σ-ideal M of microscopic sets is considered and the proof that the pair (M,M) does not satisfy (FP) is given.

scholarly journals THE FINE STRUCTURE OF THE INTUITIONISTIC BOREL HIERARCHY

2009 ◽  
Vol 2 (1) ◽  
pp. 30-101 ◽  
Author(s):  
WIM VELDMAN
Keyword(s):  
Fine Structure ◽  
Infinite Sequence ◽  
Polish Space ◽  
Main Tool ◽  
Inductive Definition ◽  
Borel Set ◽  
Borel Sets ◽  
Continuity Principle ◽  
Borel Hierarchy ◽  
Axiom Of Countable Choice

In intuitionistic analysis, a subset of a Polish space like ℝ or ${\cal N}$ is called positively Borel if and only if it is an open subset of the space or a closed subset of the space or the result of forming either the countable union or the countable intersection of an infinite sequence of (earlier constructed) positively Borel subsets of the space. The operation of taking the complement is absent from this inductive definition, and, in fact, the complement of a positively Borel set is not always positively Borel itself (see Veldman, 2008a). The main result of Veldman (2008a) is that, assuming Brouwer's Continuity Principle and an Axiom of Countable Choice, one may prove that the hierarchy formed by the positively Borel sets is genuinely growing: every level of the hierarchy contains sets that do not occur at any lower level. The purpose of the present paper is a different one: we want to explore the truly remarkable fine structure of the hierarchy. Brouwer's Continuity Principle again is our main tool. A second axiom proposed by Brouwer, his Thesis on Bars is also used, but only incidentally.

scholarly journals DASH-type cryptochromes – solved and open questions

2020 ◽  
Vol 401 (12) ◽  
pp. 1487-1493
Author(s):  
Stephan Kiontke ◽  
Tanja Göbel ◽  
Annika Brych ◽  
Alfred Batschauer
Keyword(s):  
Dna Repair ◽  
Circadian Clock ◽  
Blue Light ◽  
Dna Repair Enzymes ◽  
Repair Enzymes ◽  
Open Questions ◽  
Essential Components ◽  
Repair Activity ◽  
Almost All ◽  
Uv Lesions

AbstractDrosophila, Arabidopsis, Synechocystis, human (DASH)-type cryptochromes (cry-DASHs) form one subclade of the cryptochrome/photolyase family (CPF). CPF members are flavoproteins that act as DNA-repair enzymes (DNA-photolyases), or as ultraviolet(UV)-A/blue light photoreceptors (cryptochromes). In mammals, cryptochromes are essential components of the circadian clock feed-back loop. Cry-DASHs are present in almost all major taxa and were initially considered as photoreceptors. Later studies demonstrated DNA-repair activity that was, however, restricted to UV-lesions in single-stranded DNA. Very recent studies, particularly on microbial organisms, substantiated photoreceptor functions of cry-DASHs suggesting that they could be transitions between photolyases and cryptochromes.

scholarly journals Estimation of the mean normal measure from flat sections

2008 ◽  
Vol 40 (01) ◽  
pp. 31-48
Author(s):  
Markus Kiderlen
Keyword(s):  
A Priori ◽  
The Other ◽  
Consistent Estimator ◽  
Random Set ◽  
Normal Measure ◽  
Mild Assumption ◽  
The Mean ◽  
Vertical Sections ◽  
Almost All

We discuss the determination of the mean normal measure of a stationary random set Z ⊂ ℝ d by taking measurements at the intersections of Z with k-dimensional planes. We show that mean normal measures of sections with vertical planes determine the mean normal measure of Z if k ≥ 3 or if k = 2 and an additional mild assumption holds. The mean normal measures of finitely many flat sections are not sufficient for this purpose. On the other hand, a discrete mean normal measure can be verified (i.e. an a priori guess can be confirmed or discarded) using mean normal measures of intersections with m suitably chosen planes when m ≥ ⌊d / k⌋ + 1. This even holds for almost all m-tuples of k-dimensional planes are viable for verification. A consistent estimator for the mean normal measure of Z, based on stereological measurements in vertical sections, is also presented.

Radiocarbon Dating and Egyptian Chronology—From the “Curve of Knowns” to Bayesian Modeling

2016 ◽  
Author(s):  
Felix Höflmayer
Keyword(s):  
Bayesian Modeling ◽  
Radiocarbon Dating ◽  
State Of The Art ◽  
Near Eastern ◽  
Open Questions ◽  
The World ◽  
Dating Method ◽  
To Come ◽  
Almost All ◽  
Egyptian Archaeology

Radiocarbon dating has become a standard dating method in archaeology almost all over the world. However, in the field of Egyptology and Near Eastern archaeology, the method is still not fully appreciated. Recent years have seen several major radiocarbon projects addressing Egyptian archaeology and chronology that have led to an intensified discussion regarding the application of radiocarbon dating within the field of Egyptology. This chapter reviews the contribution of radiocarbon dating to the discipline of Egyptology, discusses state-of-the-art applications and their impact on archaeological as well as chronological questions, and presents open questions that will be addressed in the years to come.

Domain representable Lindelöf spaces are cofinally Polish

2019 ◽  
Vol 56 (4) ◽  
pp. 523-535
Author(s):  
Vladimir V. Tkachuk
Keyword(s):  
Topological Group ◽  
Polish Space ◽  
Finite Family ◽  
Locally Finite ◽  
Compact Spaces ◽  
Open Questions

Abstract We prove that, for any cofinally Polish space X, every locally finite family of non-empty open subsets of X is countable. It is also established that Lindelöf domain representable spaces are cofinally Polish and domain representability coincides with subcompactness in the class of σ-compact spaces. It turns out that, for a topological group G whose space has the Lindelöf Σ-property, the space G is domain representable if and only if it is Čech-complete. Our results solve several published open questions.

scholarly journals Study on Children’s Health Program and Maternity Health Program, Bulgaria

2019 ◽  
Vol 29 (Supplement_4) ◽  
Author(s):  
S Stefanov
Keyword(s):  
Health Program ◽  
Children's Health ◽  
Children’S Health ◽  
Insurance Fund ◽  
Face To Face ◽  
Open Questions ◽  
Doctor's Office ◽  
Almost All ◽  
The Cost ◽  
Maternity Health

Abstract Introduction The survey was conducted to establish the level access to the Children’s Health Program and the Maternal Health Program of the National Health Insurance Fund by marginalized groups. The survey was conducted in the period September-December, 2018 and covered 315 women from Nadezhda neighbourhood (Roma community),Sliven.The methodology used is social accountability and legal empowerment. Methods The survey was conducted through a face-to-face survey. A questionnaire was used with closed and open questions. Consultation with those people was anonymous. We used a “cold contact’ and a “snowball’ method. Results Almost all respondents - 97.7% - have a GP.All (who have the contact of their GP) can easily contact their doctor, although only 7 (2.3%) have the phone number of their GPs. Almost 90% pay part of the cost of treatment, few are cases of full payment or no payment.Quite low - 40.9% - are the rates of visits to the GP in the first month after birth and the appointed (mandatory and due) examinations. Only 45% were examined in the first month after birth by the testimonies of the respondents. It is imperative to take steps to raise the percentage of mothers who visit doctor’s office and receive an examinations.74.2% say they adhere to the children’s food-hygiene regime. The remaining 20.95% point out the lack of funds and the poor living conditions in the neighborhood (noise, stopping water, etc.) as reasons for not adhering to the regime. While only about 20% claim to have faced discriminatory treatment (just over 30% say they have not experienced it, and nearly half do not respond), there are indicative responses to discriminatory treatment - division in the maternity ward, offensive speech, etc. Conclusions The study achieves its goals - to provide an adequate picture of maternal access to prenatal and pediatric medical care, as well as the difficulties to ensure maximum care for children. Key messages Roma children are not treated equally under the Children’s Health. Roma mothers do not have equal access to the Maternity Health Program.

scholarly journals Stationary countable dense random sets

2000 ◽  
Vol 32 (01) ◽  
pp. 86-100 ◽  
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Probability Theory ◽  
Polish Space ◽  
Measurable Subset ◽  
Locally Compact Group ◽  
Random Sets ◽  
Random Set ◽  
Locally Compact ◽  
Polish Spaces ◽  
Probability Zero ◽  
Group Of Symmetries

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-stationarity (null-events stay null under the group action). In passing, it is noted that the property of being countable does not correspond to a measurable subset of the space of subsets of an uncountably infinite Polish space.

Sign in / Sign up

Export Citation Format

Share Document