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 ?
Therefore they have the same strong topological structures.
First, we study the strong topological structures and strong completeness of 1dimensional PN spaces.
We give the following classification for strong topological structures on 1- dimensional PN spaces.
Next, we study the strong topological structures on general n-dimensional PN spaces by using results obtained for 1-dimensional PN spaces.
We give the following classification for the strong topological structures on n- dimensional PN spaces.
However, the strong topological structure on a non-topological vector PN space has been studied scarcely in previous literature.
According to the comparison of subapertures selection methods and three topological structures, we had found that once we know the subaperture selection, the computational cost is only related to the target characteristics (the topological structure and the sparsity ratio).
During the research, we find that the selection of subaperture, topological structure and sparsity ratio of the target affect the computational cost of SA method significantly.
Note that, in fact, the topological structure of the target will affect the number of pixels of the ROI, which will be discussed in the next subsection.
To describe how much the topological structure affects the computational cost, we denote [XI] as the ROI search efficiency, which means the ratio of the number of pixels of the target area to that of the ROI area when the resolution is doubled in recursion.
Note that the topological structure of the same target may change under different resolution conditions.

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