|
Sierpiński Set
In mathematics, a Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axioms of ZFC. showed that they exist if the continuum hypothesis is true. On the other hand, they do not exist if Martin's axiom for ℵ1 is true. Sierpiński sets are weakly Luzin sets but are not Luzin sets . Example of a Sierpiński set Choose a collection of 2ℵ0 measure-0 subsets of R such that every measure-0 subset is contained in one of them. By the continuum hypothesis, it is possible to enumerate them as ''S''''α'' for countable ordinals ''α''. For each countable ordinal ''β'' choose a real number ''x''''β'' that is not in any of the sets ''S''''α'' for ''α'' < ''β'', which is possible as the union of these sets has measure 0 so is not the whole of R. Then the uncountable set ''X'' of all these real numbers ''x''''β'' has only a countable num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers. Characterizations There are many equivalent characterizations of uncountability. A set ''X'' is uncountable if and only if any of the following conditions hold: * There is no injective function (hence no bijection) from ''X'' to the set of natural numbers. * ''X'' is nonempty and for every ω-sequence of elements of ''X'', there exists at least one element of X not included in it. That is, ''X'' is nonempty and there is no surjective function from the natural numbers to ''X''. * The cardinality of ''X'' is neither finite nor equal to \aleph_0 (aleph-null, the cardinality of the natural numbers). * The set ''X'' has cardinality strictly greater than \aleph_0. The first three of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Vector Space
Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) * ''Real'' (Belinda Carlisle album) (1993) * ''Real'' (Gorgon City EP) (2013) * ''Real'' (IU EP) (2010) * ''Real'' (Ivy Queen album) (2004) * ''Real'' (Mika Nakashima album) (2013) * ''Real'' (Ednita Nazario album) (2007) * ''Real'' (Jodie Resther album), a 2000 album by Jodie Resther * ''Real'' (Michael Sweet album) (1995) * ''Real'' (The Word Alive album) (2014) * ''Real'', a 2002 album by Israel Houghton recording as Israel & New Breed Songs * "Real" (Goo Goo Dolls song) (2008) * "Real" (Gorgon City song) (2013) * "Real" (Plumb song) (2004) * "Real" (Vivid song) (2012) * "Real" (James Wesley song) (2010) * "Real", a song by Kendrick Lamar from ''Good Kid, M.A.A.D City'' * "Real", a song by NF from ''Therapy Session'' * "Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure-zero Set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = (X, \Sigma, \mu) a null set is a set S\in\Sigma such that \mu(S) = 0. Example Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers and the set of rational numbers are both countably infinite and therefore are null sets when considered as subsets of the real numbers. The Cantor set is an example of an uncountable null set. Definition Suppose A is a subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term '' the continuum'' for the real numbers. History Cantor believed the continuum hypothesis to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin's Axiom
In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, \mathfrak c, behave roughly like \aleph_0. The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments. Statement For any cardinal 𝛋, we define a statement, denoted by MA(𝛋): For any partial order ''P'' satisfying the countable chain condition (hereafter ccc) and any family ''D'' of dense sets in ''P'' such that '', D, '' ≤ 𝛋, there is a filter ''F'' on ''P'' such that ''F'' ∩ ''d'' is non-empty for every ''d'' in ''D''. \operatorname(\aleph_0) is simply true — this is known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luzin Set
In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 space without isolated points in which every nowhere-dense subset is countable. There are many minor variations of this definition in use: the T1 condition can be replaced by T2 or T3, and some authors allow a countable or even arbitrary number of isolated points. The existence of a Luzin space is independent of the axioms of ZFC. showed that the continuum hypothesis implies that a Luzin space exists. showed that assuming Martin's axiom and the negation of the continuum hypothesis, there are no Hausdorff Luzin spaces. In real analysis In real analysis and descriptive set theory, a Luzin set (or Lusin set), is defined as an uncountable subset of the reals such that every uncountable subset of is nonmeager; that is, of second Baire category. Equivalently, is an uncountable set of reals that meets every first category set in only countably many points. Luzin proved t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real-closed Field
In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. Definitions A real closed field is a field ''F'' in which any of the following equivalent conditions is true: #''F'' is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in ''F'' if and only if it is true in the reals. #There is a total order on ''F'' making it an ordered field such that, in this ordering, every positive element of ''F'' has a square root in ''F'' and any polynomial of odd degree with coefficients in ''F'' has at least one root in ''F''. #''F'' is a formally real field such that every polynomial of odd degree with coefficients in ''F'' has at least one root in ''F'', and for every element ''a'' of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamenta Mathematicae
''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems. Originally it only covered topology, set theory, and foundations of mathematics: it was the first specialized journal in the field of mathematics..... It is published by the Mathematics Institute of the Polish Academy of Sciences. History The journal was conceived by Zygmunt Janiszewski as a means to foster mathematical research in Poland.According to and to the introduction to the 100th volume of the journal (1978, pp=1–2). These two sources cite an article written by Janiszewski himself in 1918 and titled "''On the needs of Mathematics in Poland''". Janiszewski required that, in order to achieve its goal, the journal should not force Polish mathematicians to submit articles written exclusively in Polish, and should be devoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Measure Theory")