topological space

n (Mathematics) maths a set S with an associated family of subsets τ that is closed under set union and finite intersection. S and the empty set are members of τ

Thesaurus

Noun

topological space - (mathematics) any set of points that satisfy a set of postulates of some kind; "assume that the topological space is finite dimensional"mathematical spaceinfinite, space - the unlimited expanse in which everything is located; "they tested his ability to locate objects in space"; "the boundless regions of the infinite"math, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangementset - (mathematics) an abstract collection of numbers or symbols; "the set of prime numbers is infinite"subspace - a space that is contained within another spacenull space - a space that contains no points; and empty spacemanifold - a set of points such as those of a closed surface or an analogue in three or more dimensionsmetric space - a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequality

Translations

Topological Space

topological space

[¦täp·ə¦läj·ə·kəl ′spās] (mathematics) A set endowed with a topology.

Topological Space

a set among whose elements limit relations are defined in some way. Such sets may be formed by elements of any kind.

The limit relations whose existence makes a given set X a topological space consist in the following: each subset A of X has a closure [A], which consists of the elements of A and the limit points of A. In general, if a set is a topological space, its elements are called points regardless of their actual nature. We speak of assigning a topology to a given set X when we indicate in some way the closure [A] for each subset A of X. The points of [A] are said to be adherent to A.

Any metric space can be made into a topological space in a natural way. For this reason, but with some inaccuracy, a metric space is said to be a special case of a topological space. In particular, the number line, Euclidean space of any number of dimensions, and various function spaces are examples of metric spaces and, therefore, of topological spaces.

There are many ways of assigning a topology to a given set X, that is, of making the set a topological space. In the case of metric spaces, for example, the topology is assigned by means of the auxiliary concept of distance. In many cases a topology can be assigned to a given set X by means of the concept of neighborhood: for each element, or point, of X some subsets of X are identified as neighborhoods of the point; assuming the neighborhoods have been defined, we say the point x is adherent to A if each neighborhood of x contains at least one point of A.

topological space

topological space

topological space

Topological Space

topological space

Topological Space

topological space

Synonyms for topological space

noun (mathematics) any set of points that satisfy a set of postulates of some kind

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