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Intopology, adiscrete space is a particularly simple example of atopological space or similar structure, one in which the points form adiscontinuous sequence, meaning they areisolated from each other in a certain sense. The discrete topology is thefinest topology that can be given on a set. Every subset isopen in the discrete topology so that in particular, everysingleton subset is anopen set in the discrete topology.

Given a set${\displaystyle X}$:

A metric space${\displaystyle (E,d)}$ is said to beuniformly discrete if there exists apacking radius${\displaystyle r>0}$ such that, for any${\displaystyle x,y\in E,}$ one has either${\displaystyle x=y}$ or${\displaystyle d(x,y)>r.}$^[1] The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set${\displaystyle \left\{2^{-n}:n\in {\mathbb{N}}_0\right\}.}$

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${\displaystyle X=\{n^{-1}:n\in \mathbb{N}\}}$ (with metric inherited from thereal line and given by${\displaystyle d(x,y)=\left|x-y\right|}$).This is not the discrete metric; also, this space is notcomplete and hence not discrete as a uniform space.Nevertheless, it is discrete as a topological space.We say that${\displaystyle X}$ istopologically discrete but notuniformly discrete ormetrically discrete.

Additionally:

• Thetopological dimension of a discrete space is equal to 0.
• A topological space is discrete if and only if itssingletons are open, which is the case if and only if it does not contain anyaccumulation points.
• The singletons form abasis for the discrete topology.
• A uniform space${\displaystyle X}$ is discrete if and only if the diagonal${\displaystyle \{(x,x):x\in X\}}$ is anentourage.
• Every discrete topological space satisfies each of theseparation axioms; in particular, every discrete space isHausdorff, that is, separated.
• A discrete space iscompactif and only if it isfinite.
• Every discrete uniform or metric space iscomplete.
• Combining the above two facts, every discrete uniform or metric space istotally bounded if and only if it is finite.
• Every discrete metric space isbounded.
• Every discrete space isfirst-countable; it is moreoversecond-countable if and only if it iscountable.
• Every discrete space istotally disconnected.
• Every non-empty discrete space issecond category.
• Any two discrete spaces with the samecardinality arehomeomorphic.
• Every discrete space is metrizable (by the discrete metric).
• A finite space is metrizable only if it is discrete.
• If${\displaystyle X}$ is a topological space and${\displaystyle Y}$ is a set carrying the discrete topology, then${\displaystyle X}$ is evenly covered by${\displaystyle X\times Y}$ (the projection map is the desired covering)
• Thesubspace topology on theintegers as a subspace of thereal line is the discrete topology.
• A discrete space is separable if and only if it is countable.
• Any topological subspace of${\displaystyle \mathbb{R}}$ (with its usualEuclidean topology) that is discrete is necessarilycountable.^[2]

Any function from a discrete topological space to another topological space iscontinuous, and any function from a discrete uniform space to another uniform space isuniformly continuous. That is, the discrete space${\displaystyle X}$ isfree on the set${\displaystyle X}$ in thecategory 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 themorphisms. 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 metricstructure, only the uniform or topological structure. Categories more relevant to the metric structure can be found by limiting the morphisms toLipschitz continuous maps or toshort maps; however, these categories don't have free objects (on more than one element). However, the discrete metric space is free in the category ofbounded 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${\displaystyle f}$ from a topological space${\displaystyle Y}$ to a discrete space${\displaystyle X}$ is continuous if and only if it islocally constant in the sense that every point in${\displaystyle Y}$ has aneighborhood on which${\displaystyle f}$ is constant.

Everyultrafilter${\displaystyle \mathcal{U}}$ on a non-empty set${\displaystyle X}$ can be associated with a topology${\displaystyle \tau =\mathcal{U}\cup \left\{\varnothing \right\}}$ on${\displaystyle X}$ with the property thatevery non-empty proper subset${\displaystyle S}$ of${\displaystyle X}$ iseither anopen subset or else aclosed subset, but never both. Said differently,every subset is openor closed but (in contrast to the discrete topology) theonly subsets that areboth open and closed (i.e.clopen) are${\displaystyle \varnothing }$ and${\displaystyle X}$. In comparison,every subset of${\displaystyle X}$ is openand 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, anygroup can be considered as atopological 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 withPontryagin duality. A 0-dimensionalmanifold (or differentiable or analytic manifold) is nothing but a discrete and countable topological space (an uncountable discrete space is not second-countable). We can therefore view any discrete countable group as a 0-dimensionalLie group.

Aproduct ofcountably infinite copies of the discrete space ofnatural numbers ishomeomorphic to the space ofirrational numbers, with the homeomorphism given by thecontinued fraction expansion. A product of countably infinite copies of the discrete space${\displaystyle \{0,1\}}$ is homeomorphic to theCantor set; and in factuniformly homeomorphic to the Cantor set if we use theproduct uniformity on the product. Such a homeomorphism is given by usingternary notation of numbers. (SeeCantor space.) Everyfiber of alocally injective function is necessarily a discrete subspace of itsdomain.

In thefoundations of mathematics, the study ofcompactness properties of products of${\displaystyle \{0,1\}}$ is central to the topological approach to theultrafilter lemma (equivalently, theBoolean prime ideal theorem), which is a weak form of theaxiom of choice.

In some ways, the opposite of the discrete topology is thetrivial topology (also called theindiscrete topology), which has the fewest possible open sets (just theempty set and the space itself). Where the discrete topology is initial or free, the indiscrete topology is final orcofree: every functionfrom a topological spaceto an indiscrete space is continuous, etc.

• Cylinder set
• List of topologies
• Taxicab geometry

• 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.

• General topology
• Metric spaces
• Topological spaces
• Topology

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