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In topology, a **discrete space** is a particularly simple example of a topological space or similar structure, one in which the points form a *discontinuous sequence*, meaning they are * isolated * from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.

Given a set *X*:

- the
**discrete topology**on*X*is defined by letting every subset of*X*be open (and hence also closed), 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. - the
**discrete metric**on*X*is defined by**discrete metric space**or a**space of isolated points**. - a set
*S*is**discrete**in a metric space , for , if for every , there exists some (depending on ) such that for all ; such a set consists of isolated points. A set*S*is**uniformly discrete**in the metric space , for , if there exists*ε*> 0 such that for any two distinct , .

A metric space is said to be * uniformly discrete * if there exists a "packing radius" such that, for any , one has either or .^{ [1] } 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.

Proof that a discrete space is not necessarily uniformly discrete |
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Let However, Since there is always an |

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*|). This is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space. Nevertheless, it is discrete as a topological space. We say that *X* is *topologically discrete* but not *uniformly discrete* or *metrically discrete*.

Additionally:

- The topological dimension of a discrete space is equal to 0.
- 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.
- The singletons form a basis for the discrete topology.
- A uniform space
*X*is discrete if and only if the diagonal {(*x*,*x*) :*x*is in*X*} is an entourage. - Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space is Hausdorff, that is, separated.
- A discrete space is compact if and only if it is finite.
- Every discrete uniform or metric space is complete.
- Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it is finite.
- Every discrete metric space is bounded.
- 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).
- A finite space is metrizable only if it is discrete.
- If
*X*is a topological space and*Y*is a set carrying the discrete topology, then*X*is evenly covered by*X*×*Y*(the projection map is the desired covering) - The subspace topology on the integers as a subspace of the real line is the discrete topology.
- A discrete space is separable if and only if it is countable.
- Any topological subspace of ℝ (with its usual Euclidean topology) that is discrete is necessarily countable.
^{ [2] }

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. 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. 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. 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. 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). 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. 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 if it is * locally constant * in the sense that every point in *Y* has a neighborhood on which *f* is constant.

Every ultrafilter on a non-empty set can be associated with a topology on with the property that *every* non-empty proper subset of is *either* an open subset or else a closed subset, but never both. Said differently, *every* subset is open or closed but (in contrast to the discrete topology) the *only* subsets that are *both* open and closed (i.e. clopen) are and In comparison, *every* subset of is open and closed in the discrete topology.

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. Indeed, analysts may refer to the ordinary, non-topological groups studied by algebraists as "discrete groups" . In some cases, this can be usefully applied, for example in combination with Pontryagin duality. A 0-dimensional manifold (or differentiable or analytic manifold) is nothing but a discrete topological space. We can therefore view any discrete group as a 0-dimensional Lie group.

A product of countably infinite 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. A product of countably infinite 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. Such a homeomorphism is given by using ternary notation of numbers. (See Cantor space.)

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.

In some ways, the opposite of the discrete topology is the trivial topology (also called the *indiscrete topology*), which has the fewest possible open sets (just the empty set and the space itself). 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.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In mathematics, a **continuous function** is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be *discontinuous*. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

In mathematical analysis, a metric space M is called **complete** if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.

In mathematics, a **metric space** is a set together with a metric on the set. The metric is a function that defines a concept of *distance* between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In mathematics, a **topological space** is, roughly speaking, a geometrical space in which *closeness* is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In the mathematical field of topology, a **uniform space** is a set with a **uniform structure**. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis.

In mathematics, a **topological group** is a group G together with a topology on G such that both the group's binary operation and the function mapping group elements to their respective inverses are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In topology, a topological space with the **trivial topology** is one where the only open sets are the empty set and the entire space. Such spaces are commonly called **indiscrete**, **anti-discrete**, or **codiscrete**. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.

In mathematical analysis, a family of functions is **equicontinuous** if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus *sequences* of functions.

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In topology and related branches of mathematics, **total-boundedness** is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed "size"

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

In mathematics, a topological space *X* is said to be **limit point compact** or **weakly countably compact** if every infinite subset of *X* has a limit point in *X*. This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.

In topology and related areas of mathematics, a subset *A* of a topological space *X* is called **dense** if every point *x* in *X* either belongs to *A* or is a limit point of *A*; that is, the closure of *A* constitutes the whole set *X*. Informally, for every point in *X*, the point is either in *A* or arbitrarily "close" to a member of *A* — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

In functional analysis and related areas of mathematics, a **metrizable** topological vector spaces (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

- ↑ Pleasants, Peter A.B. (2000). "Designer quasicrystals: Cut-and-project sets with pre-assigned properties". In Baake, Michael (ed.).
*Directions in mathematical quasicrystals*. CRM Monograph Series.**13**. Providence, RI: American Mathematical Society. pp. 95–141. ISBN 0-8218-2629-8. Zbl 0982.52018. - ↑ Wilansky 2008, p. 35.

- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978).
*Counterexamples in Topology*(2nd ed.). Berlin, New York: Springer-Verlag. ISBN 3-540-90312-7. MR 0507446. Zbl 0386.54001. - Wilansky, Albert (17 October 2008) [1970].
*Topology for Analysis*. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-46903-4. OCLC 227923899.

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