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) Let X be an excluded point topological space with excluded point e, and let [parallel]P[parallel] = A be any subset of X.
Medhat, "Rough set theory for topological spaces," International Journal of Approximate Reasoning, vol.
Definition 6 [6] In topological space (Eq.) , a set which cannot be expressed as the union of two semi separated sets is said to be a semi connected set.
A topological space (X,[[tau].sub.[alpha]]) is [alpha]-[tau]-connected if and only if one non-empty subset which is both [alpha]-open and [alpha]-closed in X is X itself.
Let (L, [T.sub.F]) be the filter topological space. If J [member of] J satisfies [T.sup.c.sub.F] \ {L} [subset or equal to] J, then [T.sup.*.sub.F] = [J.sup.c] [union] {[empty set]}.
Generalized topological spaces are important generalizations of topological spaces and many results have been obtained by many topologist [3,4,5,11,12].
Let X and Y be topological spaces. Then a surjective function g: X [right arrow] Y is quasi sg-closed if and only if g(V)\g(X \V) is open in Y whenever V is sg-open in X.
Let S be a subset of a topological space (X, [tau]) and [m.sub.X] = [tau] (resp.
The topological space [0,[omega]] is compact, regular.
The pair (X, T) is a topological space. Topology members are called T-open or open with regard to topology T.
A topological space X is said to be triangulable if it is homeomorphic to the underlying space of some simplicial complex.
The pair (R, T) is called a neutrosophic topological space (NTS for short).

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