Discrete space - Citizendia

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology, a branch of Mathematics, a point x of a set S is called an isolated point,if there exists a neighborhood of

1 Definitions
2 Properties
3 Uses
4 Indiscrete spaces
5 Quotation
6 See also
7 References

Definitions

Given a set X:

the discrete topology on X is defined by letting every subset of X be open, and X is a discrete topological space if it is equipped with its discrete topology;
the discrete uniformity on X is defined by letting every superset of the diagonal {(x,x) : x is in X} in X × X be an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In the Mathematical field of Topology, a uniform space is a set with a uniform structure.
the discrete metric $ρ$ on X is defined by

$\rho(x,y) = \left\{\begin{matrix} 1 &\mbox{if}\ x\neq y , \\ 0 &\mbox{if}\ x = y \end{matrix}\right.$

for any $x,y \in X$ . In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In this case $(X,ρ)$ is called a discrete metric space or a space of isolated points. In Topology, a branch of Mathematics, a point x of a set S is called an isolated point,if there exists a neighborhood of

A metric space $(E, d)$ is said to be uniformly discrete if there exists $r > 0$ such that, for any $x,y \in E$ , one has either $x = y$ or $d (x, y) > r$ . The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set {1, 1/2, 1/4, 1/8, . . . } of real numbers.

Properties

The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one another. On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; an example is the metric space X := {1/n : n = 1,2,3,. . . } (with metric inherited from the real line and given by d(x,y) = |x − y|). In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has Nevertheless, it is discrete as a topological space. We say that X is topologically discrete but not uniformly discrete or metrically discrete.

Additionally:

A topological space is discrete if and only if its singletons are open, which is the case if and only if it doesn't contain any accumulation points. In Mathematics, a singleton is a set with exactly one element In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated"
The singletons form a basis for the discrete topology. In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets
A uniform space X is discrete if and only if the diagonal {(x,x) : x is in X} is an entourage. In the Mathematical field of Topology, a uniform space is a set with a uniform structure.
Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space is Hausdorff, that is, separated. In Topology and related fields of Mathematics, there are several restrictions that one often makes on the kinds of Topological spaces that one wishes to consider In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space
A discrete space is compact if and only if it is finite. ↔ In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2.
Every discrete uniform or metric space is complete. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has
Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it is finite. In Topology and related branches of Mathematics, a totally bounded space is a space that can be covered by finitely many Subsets of any
Every discrete metric space is bounded. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined
Every discrete space is first-countable, and a discrete space is second-countable if and only if it is countable. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability "
Every discrete space is totally disconnected. In Topology and related branches of Mathematics, a totally disconnected space is a Topological space which is maximally disconnected in the sense that
Every non-empty discrete space is second category. In the mathematical fields of General topology and Descriptive set theory, a meagre set (also called a meager set or a set of first category)
Any two discrete spaces with the same cardinality are homeomorphic. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" Topological equivalence redirects here see also Topological equivalence (dynamical systems.

Any function from a discrete topological space to another topological space is continuous, and any function from a discrete uniform space to another uniform space is uniformly continuous. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematical analysis, a function f ( x) is called uniformly continuous if roughly speaking small changes in the input x effect That is, the discrete space X is free on the set X in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformly continuous maps. In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets These facts are examples of a much broader phenomenon, in which discrete structures are usually free on sets.

With metric spaces, things are more complicated, because there are several categories of metric spaces, depending on what is chosen for the morphisms. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and Certainly the discrete metric space is free when the morphisms are all uniformly continuous maps or all continuous maps, but this says nothing interesting about the metric structure, only the uniform or topological structure. In Mathematics, a structure on a set, or more generally a type, consists of additional Mathematical objects that in some manner attach to the Categories more relevant to the metric structure can be found by limiting the morphisms to Lipschitz continuous maps or to short maps; however, these categories don't have free objects (on more than one element). In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions In the mathematical theory of Metric spaces a metric map or short map is a Continuous function between metric spaces that does not increase any However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined That is, any function from a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discrete metric space to another metric space bounded by 1 is short.

Going the other direction, a function f from a topological space Y to a discrete space X is continuous if and only it if is locally constant in the sense that every point in Y has a neighborhood on which f is constant. In Mathematics, a function f from a Topological space A to a set B is called locally constant, Iff In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.

Uses

A discrete structure is often used as the "default structure" on a set that doesn't carry any other natural topology, uniformity, or metric; discrete structures can often be used as "extreme" examples to test particular suppositions. For example, any group can be considered as a topological group by giving it the discrete topology, implying that theorems about topological groups apply to all groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Indeed, analysts may refer to the ordinary, non-topological groups studied by algebraists as "discrete groups" . In Mathematics, a discrete group is a group G equipped with the Discrete topology. In some cases, this can be usefully applied, for example combined with Pontryagin duality. In Mathematics, in particular in Harmonic analysis and the theory of Topological groups Pontryagin duality explains the general properties of the Fourier A 0-dimensional manifold (or differentiable or analytical manifold) is nothing but a discrete topological space. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In the spirit of the previous paragraph, we can therefore view any discrete group as a 0-dimensional Lie group. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

While discrete spaces are not very exciting from a topological viewpoint, one can easily construct interesting spaces from them. For instance, a product of countably infinitely many copies of the discrete space of natural numbers is homeomorphic to the space of irrational numbers, with the homeomorphism given by the continued fraction expansion. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} A product of countably infinitely many copies of the discrete space {0,1} is homeomorphic to the Cantor set; and in fact uniformly homeomorphic to the Cantor set if we use the product uniformity on the product. In mathematics Two has many properties in Mathematics. An Integer is called Even if it is divisible by 2 In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith In the mathematical field of Topology a uniform isomorphism or uniform homeomorphism is a special Isomorphism between Uniform spaces Such a homeomorphism is given by ternary notation of numbers. (See Cantor space. In Mathematics, the term Cantor space is sometimes used to denotethe topological abstraction of the classical Cantor set:A Topological space is aCantor )

In the foundations of mathematics, the study of compactness properties of products of {0,1} is central to the topological approach to the ultrafilter principle, which is a weak form of choice. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory In Mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given Abstract algebra. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

Indiscrete spaces

Main article: Trivial topology

In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), which has the least possible number of open sets (just the empty set and the space itself). In Topology, a Topological space with the trivial topology is one where the only Open sets are the Empty set and the entire space In Topology, a Topological space with the trivial topology is one where the only Open sets are the Empty set and the entire space In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members Where the discrete topology is initial or free, the indiscrete topology is final or cofree: every function from a topological space to an indiscrete space is continuous, etc.

Quotation

Stanislaw Ulam characterized Los Angeles, California as "a discrete space, in which there is an hour's drive between points". Stanisław Marcin Ulam ( April 13, 1909 &ndash May 13, 1984) was a Polish Mathematician who participated in the Manhattan Los Angeles (lɑˈsændʒələs los ˈaŋxeles in Spanish) is the largest City in the state of California and the American West ^[1]

References

^ Stanislaw Ulam's autobiography, Adventures of a Mathematician. In Mathematics, a cylinder set is the natural Open set of a Product topology. Stanisław Marcin Ulam ( April 13, 1909 &ndash May 13, 1984) was a Polish Mathematician who participated in the Manhattan

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