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Every discrete space is first-countable; it is moreover second-countable if and only if it is countable. Every discrete space is totally disconnected. Every non-empty discrete space is second category. Any two discrete spaces with the same cardinality are homeomorphic. Every discrete space is metrizable (by the discrete metric).
The following are examples of totally disconnected spaces: Discrete spaces; The rational numbers; The irrational numbers; The p-adic numbers; more generally, all profinite groups are totally disconnected. The Cantor set and the Cantor space; The Baire space; The Sorgenfrey line; Every Hausdorff space of small inductive dimension 0 is totally ...
If the space X is a metric space, for example a Euclidean space, then an element x of S is an isolated point of S if there exists an open ball around x that contains only finitely many elements of S. A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space).
In topology, a branch of mathematics, a discrete two-point space is the simplest example of a totally disconnected discrete space. The points can be denoted by the symbols 0 and 1. The points can be denoted by the symbols 0 and 1.
Alexandrov-discrete spaces can thus be viewed as a generalization of finite topological spaces. Due to the fact that inverse images commute with arbitrary unions and intersections, the property of being an Alexandrov-discrete space is preserved under quotients. Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov.
Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected. The simplest example is the discrete two-point space. [3] On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ring consists of two points and is connected.
Discrete space, a simple example of a topological space; Discrete spline interpolation, the discrete analog of ordinary spline interpolation; Discrete time, non-continuous time, which results in discrete-time samples; Discrete variable, non-continuous variable; Discrete pitch, a pitch with a steady frequency, rather than an indiscrete gliding ...
Every Stonean space is a Stone space, but not vice versa. In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras. An extremally disconnected first-countable collectionwise Hausdorff space must be discrete.