# Topological Space

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Related to Topological Space: metric 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.

References in periodicals archive ?
J]-closed sets in generalized topological spaces and some of their properties are established.
A function between topological spaces is semi-strongly continuous provided the preimage of each semi-open set is open.
Caldas, Semi-generalized continuous maps in topological spaces, Port.
A path in a topological space X is a continuous function f : [0, 1] [right arrow] X.
Let (X, [tau]) be a topological space of IFRSs in X and let P [member of] [C.
In the topological spaces, this notion is equivalent to the concepts such as semi-connected topological space and _D-space.
left arrow]) Let (X, [tau]) be a topological space such that for any points x and y in X, [Ker.
Note here fuzzy topological space means fuzzy topological space of Change [3].
alpha]] : [alpha] [member of] I} be topological spaces.
A subset A of a topological space (X,[tau]) is called [beta]-open if A [subset or equal to] Cl(Int(Cl(A))), where Cl(A)and Int(A) denote the closure and the interior of A, respectively.
The pair (X,[delta]) is called fuzzy topological space.
These three works map out a topological space in which various histories converge: the pre- and post-9/11 housing markets, the emigration of an Italian revolutionary to New York in 1850, the possible theft of Meucci's invention by Alexander Graham Bell, and the interdependency of the art and property markets in present-day Manhattan.

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