Topological Space

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topological space

[¦täp·ə¦läj·ə·kəl ′spās]
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 ?
2]) is also a homeomorphic mapping, then both the linear structure and the strong topological structure are the same on these two spaces.
5, the blade topological structure suggests a rough impression of the blade material distribution, which are, the cap, web and skin.
After analyzing the topological structure of the map and the size relations, small islands (e.
The largest obstacle in putting traditional GIS into GIS-T is the node-arc topological structure relied on maps[8][9].
In the second part of the proposal, I will study properties of interaction-driven electronic phases that have an underlying topological structure.
In 1985, Sostak [7] introduced a new form of topological structure.
Though the GAP tree may be computed for a source data set which has a planar partitioning topology, the GAP tree itself is not a topological structure.
The project will also look at a new discretisation technology developed at the host institution which preserves geometrical and topological structure with a view to testing this new technology in gravity applications.
The Quad-Edge data structure was introduced by Guibas and Stolfi (1985) as a primitive topological structure for the representation of any subdivision on a two-dimensional manifold (Kendig 1977, p.
The Ability to browse topological structure of models and zoom to selected entities.
Each topological structure has a carefully designed closed form formula to cover a wide range of geometry instances.
Thron, Topological structures, Holt, Rinehart and Winston, New York, 1966.

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