Topological Space


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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.

References in periodicals archive ?
1] A neutrosophic set A in a neutrosophic topological space (X, T) is called a neutrosophic [alpha]-open set (N[alpha]OS) if A [subset or equal to] N int(N cl(N int(A))).
alpha] [member of] [DELTA]} of soft sets over X is soft locally finite in the soft topological space (X, E, [?
Throughout this paper, and are always topological spaces on which no separation axioms are assumed.
alpha]]) is topological space such that the only non-empty subsets of X which is [alpha]-open as well as [alpha]-closed in X is X itself.
The topological space of the camp (and the fragments of life it makes visible) matters because, for Butler, it is from there that the struggle to reinvest topographical representation can take place.
J]-closed sets in generalized topological spaces and some of their properties are established.
Let X and Y be topological spaces and let g: X [right arrow] Y be a sg-continuous quasi sg-closed surjective function.
Semi-open sets, preopen sets, [alpha]-open sets and, [beta]-open sets play an important role in the researching of generalizations of continuity in topological spaces and bitopological spaces.
The topological space (R,T) is hereditarily Lindelof.
4 [3] Let (X, [tau]) be a topological space and (X, [[tau].
A set U of a topological space (X, T) is a neighborhood (T-neighborhood) of a point x if and only if U contains an open set to which x belongs.
An attractor is a piece of topological space that pulls objects into itself.

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