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Extremally Disconnected Space
In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries, and is sometimes mistaken by spellcheckers for the homophone ''extremely disconnected''.) An extremally disconnected space that is also compact and Hausdorff is sometimes called a Stonean space. This is not the same as a Stone space, which is a totally disconnected compact Hausdorff space. Every Stonean space is a Stone space, but not vice versa. In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras. An extremally disconnected first-countable collectionwise Hausdorff space must be discrete. In particular, for metric spaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to the property of being discrete (every set is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AW*-algebra
In mathematics, an AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951. As operator algebras, von Neumann algebras, among all C*-algebras, are typically handled using one of two means: they are the dual space of some Banach space, and they are determined to a large extent by their projections. The idea behind AW*-algebras is to forgo the former, topological, condition, and use only the latter, algebraic, condition. Definition Recall that a projection of a C*-algebra is a self-adjoint idempotent element. A C*-algebra ''A'' is an AW*-algebra if for every subset ''S'' of ''A'', the left annihilator :\mathrm_L(S)=\\, is generated as a left ideal by some projection ''p'' of ''A'', and similarly the right annihilator is generated as a right ideal by some projection ''q'': :\forall S \subseteq A\, \exists p,q \in \mathrm(A) \colon \mathrm_L(S)=Ap, \quad \mathrm_R(S)=qA. Hence an AW*-algebra is a C*-algebras that is at the sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Arhangelskii
Alexander Vladimirovich Arhangelskii (russian: Александр Владимирович Архангельский, ''Aleksandr Vladimirovich Arkhangelsky'', born 13 March 1938 in Moscow) is a Russian mathematician. His research, comprising over 200 published papers, covers various subfields of general topology. He has done particularly important work in metrizability theory and generalized metric spaces, cardinal functions, topological function spaces and other topological groups, and special classes of topological maps. After a long and distinguished career at Moscow State University, he moved to the United States in the 1990s. In 1993 he joined the faculty of Ohio University, from which he retired in 2011. Biography Arhangelskii was the son of Vladimir Alexandrovich Arhangelskii and Maria Pavlova Radimova, who divorced by the time he was four years old. He was raised in Moscow by his father. He was also close to his uncle, childless aircraft designer Alexander Arkhangelsky ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Disconnected Space
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the ''only'' connected proper subsets. An important example of a totally disconnected space is the Cantor set, which is homeomorphic to the set of ''p''-adic integers. Another example, playing a key role in algebraic number theory, is the field of ''p''-adic numbers. Definition A topological space X is totally disconnected if the connected components in X are the one-point sets. Analogously, a topological space X is totally path-disconnected if all path-components in X are the one-point sets. Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons. That is, a topological space X is totally separated space if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz–Markov–Kakutani Representation Theorem
In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuous functions on the unit interval, who extended the result to some non-compact spaces, and who extended the result to compact Hausdorff spaces. There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures. The representation theorem for positive linear functionals on ''Cc''(''X'') The following theorem represents positive linear functionals on ''Cc''(''X''), the space of conti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Retract (topology)
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of ''continuously shrinking'' a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex. Definitions Retract Let ''X'' be a topological space and ''A'' a subspace of ''X''. Then a continuous map :r\colon X \to A is a retraction if the restriction of ''r'' to ''A'' is the identity map on ''A''; that is, r(a) = a for all ''a'' in ''A''. Equivalently, denoting by :\iota\colon A \hookrightarrow X the inclusion, a retraction is a continuous map ''r'' such that :r \circ \iota = \operatorname_A, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Object
In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object. Definition An object P in a category \mathcal is ''projective'' if for any epimorphism e:E\twoheadrightarrow X and morphism f:P\to X, there is a morphism \overline:P\to E such that e\circ \overline=f, i.e. the following diagram commutes: That is, every morphism P\to X factors through every epimorphism E\twoheadrightarrow X. If ''C'' is locally small, i.e., in particular \operatorname_C(P, X) is a set for any object ''X'' in ''C'', this definition is equivalent to the condition that the hom functor (also known as corepresentable functor) : \operatorname(P,-)\colon\mathcal\to\mathbf preserves epimorphisms. Projective objects in abelian categories If the category ''C'' is an abelian category such as, for example, the category of abel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: :\. Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set is called finite if there exists a bijection :f\colon S\to\ for some natural number . The number is the set's cardinality, denoted as . The empty set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (topology)
In mathematics, a base (or basis) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under some c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperconnected Space
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is preferred in algebraic geometry. For a topological space ''X'' the following conditions are equivalent: * No two nonempty open sets are disjoint. * ''X'' cannot be written as the union of two proper closed sets. * Every nonempty open set is dense in ''X''. * The interior of every proper closed set is empty. * Every subset is dense or nowhere dense in ''X''. * No two points can be separated by disjoint neighbourhoods. A space which satisfies any one of these conditions is called ''hyperconnected'' or ''irreducible''. Due to the condition about neighborhoods of distinct points being in a sense the opposite of the Hausdorff property, some authors call such spaces anti-Hausdorff. An irreducible set is a subset of a topological space for wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
