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Morita Conjectures
The Morita conjectures in general topology are certain problems about normal spaces, now solved in the affirmative. The conjectures, formulated by Kiiti Morita in 1976, asked # If X \times Y is normal for every normal space ''Y'', is ''X'' a discrete space? # If X \times Y is normal for every normal P-space#P-spaces in the sense of Morita, P-space ''Y'', is ''X'' metrizable? # If X \times Y is normal for every normal countably paracompact space ''Y'', is ''X'' metrizable and sigma-locally compact? The answers were believed to be affirmative. Here a normal P-space ''Y'' is characterised by the property that the product with every metrizable ''X'' is normal; thus the conjecture was that the converse holds. Keiko Chiba, Teodor C. Przymusiński, and Mary Ellen Rudin proved conjecture (1) and showed that conjectures (2) and (3) cannot be proven false under the standard Zermelo–Fraenkel set theory, ZFC axioms for mathematics (specifically, that the conjectures hold under the axiom o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': * Continuous functions, intuitively, take nearby points to nearby points. * Compact sets are those that can be covered by finitely many sets of arbitrarily small size. * Connected sets are sets that cannot be divided into two pieces that are far apart. The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces. Definitions A topological space ''X'' is a normal space if, given any disjoint closed sets ''E'' and ''F'', there are neighbourhoods ''U'' of ''E'' and ''V'' of ''F'' that are also disjoint. More intuitively, this condition says that ''E'' and ''F'' can be separated by neighbourhoods. A T4 space is a T1 space ''X'' that is normal; this is equivalent to ''X'' being normal and Hausdorff. A completely normal space, or , is a topological space ''X'' such that every subspace of ''X'' with subspace topology is a normal space. It turns out that ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kiiti Morita
was a Japanese mathematician working in algebra and topology. Morita was born in 1915 in Hamamatsu, Shizuoka Prefecture and graduated from the Tokyo Higher Normal School in 1936. Three years later he was appointed assistant at the Tokyo University of Science. He received his Ph.D. from Osaka University in 1950, with a thesis in topology. After teaching at the Tokyo Higher Normal School, he became professor at the University of Tsukuba in 1951. He held this position until 1978, after which he taught at Sophia University. Morita died of heart failure in 1995 at the Sakakibara Heart Institute in Tokyo; he was survived by his wife, Tomiko, his son, Yasuhiro, and a grandson. He introduced the concepts now known as Morita equivalence and Morita duality which were given wide circulation in the 1960s by Hyman Bass in a series of lectures. The Morita conjectures The Morita conjectures in general topology are certain problems about normal spaces, now solved in the affirmative. The conjec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be '' uniformly discrete'' if there exists a ' r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left\. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-space
In the mathematical field of topology, there are various notions of a ''P''-space and of a ''p''-space. Generic use The expression ''P-space'' might be used generically to denote a topological space satisfying some given and previously introduced topological invariant ''P''. This might apply also to spaces of a different kind, i.e. non-topological spaces with additional structure. ''P-spaces'' in the sense of Gillman–Henriksen A ''P-space'' in the sense of Gillman– Henriksen is a topological space in which every countable intersection of open sets is open. An equivalent condition is that countable unions of closed sets are closed. In other words, Gδ sets are open and Fσ sets are closed. The letter ''P'' stands for both ''pseudo-discrete'' and ''prime''. Gillman and Henriksen also define a ''P-point'' as a point at which any prime ideal of the ring of real-valued continuous functions is maximal, and a P-space is a space in which every point is a P-point. Different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) such that the topology induced by d is \mathcal. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Properties Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic. Metrization theorems One of the first widely recognized metrization theorems was . This states that every H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact. Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. Howeve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mary Ellen Rudin
Mary Ellen Rudin (December 7, 1924 – March 18, 2013) was an American mathematician known for her work in set-theoretic topology. In 2013, Elsevier established the Mary Ellen Rudin Young Researcher Award, which is awarded annually to a young researcher, mainly in fields adjacent to general topology. Early life and education Mary Ellen (Estill) Rudin was born in Hillsboro, Texas to Joe Jefferson Estill and Irene (Shook) Estill. Her mother Irene was an English teacher before marriage, and her father Joe was a civil engineer. The family moved with her father's work, but spent a great deal of Mary Ellen's childhood around Leakey, Texas.Albers, D.J. and Reid, C. (1988) "An Interview with Mary Ellen Rudin". ''The College of Mathematics Journal'' 19(2) pp.114-137 She had one sibling, a younger brother. Both of Rudin's maternal grandmothers had attended Mary Sharp College near their hometown of Winchester, Tennessee. Rudin remarks on this legacy and how much her family valued educat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible universe, constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann universe and the constructible universe, respectively. The axiom, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see list of large cardinal properties). Generalizations of this axiom are explored in inner model theory. Implications The axiom of constructibility implies the axiom of choice (AC), given Zermelo–Fraenkel set theory without the axiom of choice (ZF). It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the Continuum hypothesis#The generalized continuum hypothesis, generalized continuum hypothesis, the negation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
