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Scattered Space
In mathematics, a scattered space is a topological space ''X'' that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset ''A'' of ''X'' contains a point isolated in ''A''. A subset of a topological space is called a scattered set if it is a scattered space with the subspace topology. Examples * Every discrete space is scattered. * Every ordinal number with the order topology is scattered. Indeed, every nonempty subset ''A'' contains a minimum element, and that element is isolated in ''A''. * A space ''X'' with the particular point topology, in particular the Sierpinski space, is scattered. This is an example of a scattered space that is not a T1 space, T1 space. * The closure of a scattered set is not necessarily scattered. For example, in the Euclidean plane \R^2 take a countably infinite discrete set ''A'' in the unit disk, with the points getting denser and denser as one approaches the boundary. For example, take the union of the vertices of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |