topological space

topological space
a set with a collection of subsets or open sets satisfying the properties that the union of open sets is an open set, the intersection of two open sets is an open set, and the given set and the empty set are open sets.
[1945-50]

* * *

      in mathematics, generalization of Euclidean spaces in which the idea of closeness, or limits, is described in terms of relationships between sets rather than in terms of distance. Every topological space consists of: (1) a set of points; (2) a class of subsets defined axiomatically as open sets; and (3) the set operations of union and intersection. In addition, the class of open sets in (2) must be defined in such a manner that the intersection of any finite number of open sets is itself open and the union of any, possibly infinite, collection of open sets is likewise open. The concept of limit point is of fundamental importance in topology; a point p is called a limit point of the set S if every open set containing p also contains some point (s) of S (points other than p, should p happen to lie in S ). The concept of limit point is so basic to topology that, by itself, it can be used axiomatically to define a topological space by specifying limit points for each set according to rules known as the Kuratowski closure axioms. Any set of objects can be made into a topological space in various ways, but the usefulness of the concept depends on the manner in which the limit points are separated from each other. Most topological spaces that are studied have the Hausdorff (Hausdorff space) property, which states that any two points can be contained in nonoverlapping open sets, guaranteeing that a sequence of points can have no more than one limit point.

* * *


Universalium. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Topological space — Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The… …   Wikipedia

  • topological space — noun (mathematics) any set of points that satisfy a set of postulates of some kind assume that the topological space is finite dimensional • Syn: ↑mathematical space • Topics: ↑mathematics, ↑math, ↑maths …   Useful english dictionary

  • topological space — noun Date: 1926 a set with a collection of subsets satisfying the conditions that both the empty set and the set itself belong to the collection, the union of any number of the subsets is also an element of the collection, and the intersection of …   New Collegiate Dictionary

  • topological space — noun a) A set, together with a collection of its subsets that form a topology on the set. b) The set itself …   Wiktionary

  • Finite topological space — In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space for which there are only finitely many points.While topology is mostly interesting only for… …   Wikipedia

  • Noetherian topological space — In mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the… …   Wikipedia

  • Localization of a topological space — In mathematics, well behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in… …   Wikipedia

  • Space (mathematics) — This article is about mathematical structures called spaces. For space as a geometric concept, see Euclidean space. For all other uses, see space (disambiguation). A hierarchy of mathematical spaces: The inner product induces a norm. The norm… …   Wikipedia

  • Topological manifold — In mathematics, a topological manifold is a Hausdorff topological space which looks locally like Euclidean space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout… …   Wikipedia

  • Topological group — Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”