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Indecomposable Continuum
In point-set topology, an indecomposable continuum is a continuum (topology), continuum that is indecomposable, i.e. that cannot be expressed as the union of any two of its Subset#Definitions, proper subcontinua. In 1910, L. E. J. Brouwer was the first to describe an indecomposable continuum. Indecomposable continua have been used by topologists as a source of counterexamples. They also occur in dynamical systems. Definitions A ''continuum'' C is a nonempty Compact space, compact Connected space, connected metric space. The Arc (topology), arc, the n-sphere, ''n''-sphere, and the Hilbert cube are examples of Connected_space#Path_connectedness, path-connected continua; the topologist's sine curve and Shape_theory_(mathematics)#Warsaw_Circle, Warsaw circle are examples of non-path-connected continua. A ''subcontinuum'' C' of a continuum C is a closed set, closed, connected subset of C. A space is ''nondegenerate'' if it is not equal to a single point. A continuum C is ''decompos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indecomposability (constructive Mathematics)
In intuitionistic analysis and in computable analysis, indecomposability or indivisibility (german: Unzerlegbarkeit, from the adjective ''unzerlegbar'') is the principle that the continuum cannot be partitioned into two nonempty pieces. This principle was established by Brouwer in 1928 English translation of §1 see p.490–492 of: using intuitionistic principles, and can also be proven using Church's thesis. The analogous property in classical analysis is the fact that every continuous function from the continuum to is constant. It follows from the indecomposability principle that any property of real numbers that is ''decided'' (each real number either has or does not have that property) is in fact trivial (either all the real numbers have that property, or else none of them do). Conversely, if a property of real numbers is not trivial, then the property is not decided for all real numbers. This contradicts the law of the excluded middle, according to which every proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composant
In point-set topology, the composant of a point ''p'' in a continuum ''A'' is the union of all proper subcontinua of ''A'' that contain ''p''. If a continuum is indecomposable, then its composants are pairwise disjoint. The composants of a continuum are dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ... in that continuum. References * * Continuum theory {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space ''X'', the collection of all Borel sets on ''X'' forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Ternary Set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned the ternary construction only in passing, as an example of a more general idea, that of a perfect set that is nowhere dense. More generally, in topology, ''a'' Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). By a theorem of Brouwer, this is equivalent to being perfect nonempty, compact metrizable and zero dimensional. Construction and formula of the ternary set The Cantor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory. By working in the dual category, that is by reverting the arrows, an inverse limit becomes a direct limit or ''inductive limit'', and a ''limit'' becomes a colimit. Formal definition Algebraic objects We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (I, \leq) be a directed poset (not all authors require ''I'' to be directed). Let (''A''''i'')''i''∈''I'' be a family of groups and suppose we have a family of homomorphisms f_: A_j \to A_i for all i \leq j (note the order) with the following properties: # f_ is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-arc
In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. Bing proved that, in a certain well-defined sense, most continua in R''n'', ''n'' ≥ 2, are homeomorphic to the pseudo-arc. History In 1920, Bronisław Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane R2 must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in R2 that is homeomorphic to each of its nondegenerate subcontinua must be an arc. In 1922, Knaster discovered the first example of a hereditarily indecomposable continuum ''K'', later named the pseudo-arc, giving a negative answer to a Mazurkiewicz question. In 1948, R. H. Bing proved that Knaster's continuum is homogeneous, i.e. for any two of its points there is a homeomorphism tak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bronisław Knaster
Bronisław Knaster (22 May 1893 – 3 November 1980) was a Polish mathematician; from 1939 a university professor in Lwów and from 1945 in Wrocław. He is known for his work in point-set topology and in particular for his discoveries in 1922 of the hereditarily indecomposable continuum or pseudo-arc and of the Knaster continuum, or buckethandle continuum. Together with his teacher Hugo Steinhaus and his colleague Stefan Banach, he also developed the last diminisher procedure for fair cake cutting. Knaster received his Ph.D. degree from University of Warsaw in 1925, under the supervision of Stefan Mazurkiewicz. See also *Knaster–Tarski theorem *Knaster–Kuratowski fan *Knaster's condition In mathematics, a partially ordered set ''P'' is said to have Knaster's condition upwards (sometimes property (K)) if any uncountable subset ''A'' of ''P'' has an upwards-linked uncountable subset. An analogous definition applies to Knaster's con ... References 1893 births 199 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stefan Mazurkiewicz
Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability. He was a student of Wacław Sierpiński and a member of the Polish Academy of Learning (''PAU''). His students included Karol Borsuk, Bronisław Knaster, Kazimierz Kuratowski, Stanisław Saks, and Antoni Zygmund. For a time Mazurkiewicz was a professor at the University of Paris; however, he spent most of his career as a professor at the University of Warsaw. The Hahn–Mazurkiewicz theorem, a basic result on curves prompted by the phenomenon of space-filling curves, is named for Mazurkiewicz and Hans Hahn. His 1935 paper ''Sur l'existence des continus indécomposables'' is generally considered the most elegant piece of work in point-set topology. During the Polish–Soviet War (1919–21), Mazurkiewicz as early as 1919 broke the most common Russian cipher for the Polish General Staff's cryptological agency. Thanks to this, ord ... [...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 devote ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warsaw School (mathematics)
Warsaw School of Mathematics is the name given to a group of mathematicians who worked at Warsaw, Poland, in the two decades between the World Wars, especially in the fields of logic, set theory, point-set topology and real analysis. They published in the journal ''Fundamenta Mathematicae'', founded in 1920—one of the world's first specialist pure-mathematics journals. It was in this journal, in 1933, that Alfred Tarski—whose illustrious career would a few years later take him to the University of California, Berkeley—published his celebrated theorem on the undefinability of the notion of truth. Notable members of the Warsaw School of Mathematics have included: * Wacław Sierpiński * Kazimierz Kuratowski * Edward Marczewski * Bronisław Knaster * Zygmunt Janiszewski * Stefan Mazurkiewicz * Stanisław Saks * Karol Borsuk * Roman Sikorski * Nachman Aronszajn * Samuel Eilenberg Additionally, notable logicians of the Lwów–Warsaw School of Logic, working at Warsaw, have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Takeo Wada
was a Japanese mathematician at Kyoto University working in analysis and topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing .... He suggested the Lakes of Wada to Kunizo Yoneyama, who wrote about them and named them after Wada. Publications * References * * 1882 births 1944 deaths 20th-century Japanese mathematicians {{Japan-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kunizo Yoneyama
was a Japanese mathematician at Kyoto University working in topology. In 1917, he published the construction of the Lakes of Wada, which he named after his teacher Takeo Wada was a Japanese mathematician at Kyoto University working in analysis and topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deform ..., to whom he credited the discovery. Publications * * * References * * Japanese mathematicians 1877 births 1968 deaths Academic staff of Kyoto University {{Japan-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
