|
Sober Space
In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible space, irreducible closed subset of ''X'' is the closure (topology), closure of exactly one point of ''X'': that is, every nonempty irreducible closed subset has a unique generic point. Definitions Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the Kolmogorov space, T0 axiom. Replacing it with "at least one" is equivalent to the property that the T0 quotient space (topology), quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature. With irreducible closed sets A closed set is Hyperconnected space, irreducible if it cannot be written as the union of two proper closed subsets. A space is sober if every nonempty irreducible closed subset is the closure of a unique point. In terms of morphisms o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each Mathematical object, object X in ''C'' to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up To
Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, " is unique up to " means that all objects under consideration are in the same equivalence class with respect to the relation . Moreover, the equivalence relation is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation that relates two lists if one can be obtained by reordering (permutation, permuting) the other. As another example, the statement "the solution to an indefinite integral is , up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join (mathematics), join) and a unique infimum (also called a greatest lower bound or meet (mathematics), meet). An example is given by the power set of a set, partially ordered by Subset, inclusion, for which the supremum is the Union (set theory), union and the infimum is the Intersection (set theory), intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic Identity (mathematics), identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilatti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofinite Topology
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but is countable, then one says the set is cocountable. These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum. This use of the prefix "" to describe a property possessed by a set's mplement is consistent with its use in other terms such as " meagre set". Boolean algebras The set of all subsets of X that are either finite or cofinite forms a Boolean algebra, which means that it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the on X. In the other direction, a Boolean algebra A has a unique non-principal ultrafilter (that is, a maximal filter not generated by a single element of the algebra) if and only if there exists an inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T1 Space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms. Definitions Let ''X'' be a topological space and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' are if each lies in a neighbourhood that does not contain the other point. * ''X'' is called a T1 space if any two distinct points in ''X'' are separated. * ''X'' is called an R0 space if any two topologically distinguishable points in ''X'' are separated. A T1 space is also called an accessible space or a space with Fréchet topology and an R0 space is also called a symmetric space. (The term also has an entirely different meaning in functional analysis. For this reason, the term ''T1 space'' is preferred. There is also a n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparability
In mathematics, two elements ''x'' and ''y'' of a set ''P'' are said to be comparable with respect to a binary relation ≤ if at least one of ''x'' ≤ ''y'' or ''y'' ≤ ''x'' is true. They are called incomparable if they are not comparable. Rigorous definition A binary relation on a set P is by definition any subset R of P \times P. Given x, y \in P, x R y is written if and only if (x, y) \in R, in which case x is said to be to y by R. An element x \in P is said to be , or (), to an element y \in P if x R y or y R x. Often, a symbol indicating comparison, such as \,,\, \geq, and many others) is used instead of R, in which case x < y is written in place of which is why the term "comparable" is used. Comparability with respect to induces a canonical binary relation on ; specifically, the induced by is defined to be the set of all pairs such that i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T0 Space
T, or t, is the twentieth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''tee'' (pronounced ), plural ''tees''. It is derived from the Semitic Taw 𐤕 of the Phoenician and Paleo-Hebrew script (Aramaic and Hebrew Taw ת/𐡕/, Syriac Taw ܬ, and Arabic ت Tāʼ) via the Greek letter τ (tau). In English, it is most commonly used to represent the voiceless alveolar plosive, a sound it also denotes in the International Phonetic Alphabet. It is the most commonly used consonant and the second-most commonly used letter in English-language texts. History '' Taw'' was the last letter of the Western Semitic and Hebrew alphabets. The sound value of Semitic ''Taw'', the Greek alphabet Tαυ (''Tau''), Old Italic and Latin T has remained fairly constant, representing in each of these, and it has also kept its original basic shape in most of these alphabets. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T2 Space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be ''separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff spaces ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
