Editing Discrete space

{{Short description|Type of topological space}}
{{Refimprove|date=March 2011}}
In [[topology]], a '''discrete space''' is a particularly simple example of a [[topological space]] or similar structure, one in which the points form a {{em|discontinuous sequence}}, meaning they are ''[[Isolated point|isolated]]'' from each other in a certain sense. The discrete topology is the [[Comparison of topologies|finest]] topology that can be given on a set. Every subset is [[Open set|open]] in the discrete topology so that in particular, every [[Singleton (mathematics)|singleton subset]] is an [[open set]] in the discrete topology.
<!-- Dummy edit, can be removed -->

== Definitions ==
Given a set <math>X</math>:{{unordered list
| the '''{{visible anchor|discrete topology}}''' on <math>X</math> is defined by letting every [[subset]] of <math>X</math> be [[Open set|open]] (and hence also [[Closed set|closed]]), and <math>X</math> is a '''{{visible anchor|discrete topological space}}''' if it is equipped with its discrete topology;
| the '''{{visible anchor|discrete uniformity|discrete [[Uniform space|uniformity]]}}''' on <math>X</math> is defined by letting every [[superset]] of the diagonal <math>\{(x,x) : x \in X\}</math> in <math>X \times X</math> be an [[Entourage (topology)#Entourage definition|entourage]], and <math>X</math> is a '''{{visible anchor|discrete uniform space}}''' if it is equipped with its discrete uniformity.
| the '''{{visible anchor|discrete metric|text=discrete [[Metric space|metric]]}}''' <math>\rho</math> on <math>X</math> is defined by <math display=block>\rho(x,y) = \begin{cases} 
1 &\text{if}\ x\neq y , \\
0 &\text{if}\ x = y
\end{cases}</math> for any <math>x,y \in X.</math> In this case <math>(X,\rho)</math> is called a '''{{visible anchor|discrete metric space}}''' or a '''space of [[isolated point]]s'''.
| a '''{{visible anchor|discrete subspace}}''' of some given topological space <math>(Y, \tau)</math> refers to a [[topological subspace]] of <math>(Y, \tau)</math> (a subset of <math>Y</math> together with the [[subspace topology]] that <math>(Y, \tau)</math> induces on it) whose topology is equal to the discrete topology. For example, if <math>Y := \R</math> has its usual [[Euclidean topology]] then <math>S = \left\{\tfrac{1}{2}, \tfrac{1}{3}, \tfrac{1}{4}, \ldots\right\}</math> (endowed with the subspace topology) is a discrete subspace of <math>\R</math> but <math>S \cup \{0\}</math> is not. 
| a [[Set (mathematics)|set]] <math>S</math> is '''discrete''' in a [[metric space]] <math>(X,d),</math> for <math>S \subseteq X,</math> if for every <math>x \in S,</math> there exists some <math>\delta > 0</math> (depending on <math>x</math>) such that <math>d(x,y) > \delta</math> for all <math>y \in S\setminus\{x\}</math>; such a set consists of [[isolated point]]s.  A set <math>S</math> is '''uniformly discrete''' in the [[metric space]] <math>(X,d),</math> for <math>S \subseteq X,</math> if there exists <math>\varepsilon > 0</math> such that for any two distinct <math>x, y \in S, d(x, y) > \varepsilon.</math>
}}

A metric space <math>(E,d)</math> is said to be ''[[Uniformly discrete set|uniformly discrete]]'' if there exists a ''{{visible anchor|packing radius}}'' <math>r > 0</math> such that, for any <math>x,y \in E,</math> one has either <math>x = y</math> or <math>d(x,y) > r.</math><ref>{{cite book | zbl=0982.52018 | last=Pleasants | first=Peter A.B. | chapter=Designer quasicrystals: Cut-and-project sets with pre-assigned properties | editor1-last=Baake | editor1-first=Michael | title=Directions in mathematical quasicrystals | location=Providence, RI | publisher=[[American Mathematical Society]] | series=CRM Monograph Series | volume=13 | pages=95–141 | year=2000 | isbn=0-8218-2629-8 }}</ref>  The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set <math>\left\{2^{-n} : n \in \N_0\right\}.</math>

{{math proof|title=Proof that a discrete space is not necessarily uniformly discrete | proof =
Let <math display="inline">X = \left\{2^{-n} : n \in \N_0 \right\} = \left\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots\right\},</math> consider this set using the usual metric on the real numbers. Then, <math>X</math> is a discrete space, since for each point <math>x_n = 2^{-n} \in X,</math> we can surround it with the open interval <math>(x_n - \varepsilon, x_n + \varepsilon),</math> where <math>\varepsilon = \tfrac{1}{2} \left(x_n - x_{n+1}\right) = 2^{-(n+2)}.</math> The intersection <math>\left(x_n - \varepsilon, x_n + \varepsilon\right) \cap X</math> is therefore trivially the singleton <math>\{x_n\}.</math> Since the intersection of an open set of the real numbers and <math>X</math> is open for the induced topology, it follows that <math>\{x_n\}</math> is open so singletons are open and <math>X</math> is a discrete space.

However, <math>X</math> cannot be uniformly discrete. To see why, suppose there exists an <math>r > 0</math> such that <math>d(x,y) > r</math> whenever <math>x \neq y.</math> It suffices to show that there are at least two points <math>x</math> and <math>y</math> in <math>X</math> that are closer to each other than <math>r.</math> Since the distance between adjacent points <math>x_n</math> and <math>x_{n+1}</math> is <math>2^{-(n+1)},</math> we need to find an <math>n</math> that satisfies this inequality:
<math display=block>\begin{align}
               2^{-(n+1)} &< r \\
                        1 &< 2^{n+1}r \\
                   r^{-1} &< 2^{n+1} \\
\log_2\left(r^{-1}\right) &< n+1 \\
               -\log_2(r) &< n+1 \\
           -1 - \log_2(r) &< n
\end{align}</math>

Since there is always an <math>n</math> bigger than any given real number, it follows that there will always be at least two points in <math>X</math> that are closer to each other than any positive <math>r,</math> therefore <math>X</math> is not uniformly discrete.
}}

== 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 <math>X = \{n^{-1} : n \in \N\}</math> (with metric inherited from the [[real line]] and given by <math>d(x,y) = \left|x - y\right|</math>).
This is not the discrete metric; also, this space is not [[complete (topology)|complete]] and hence not discrete as a uniform space.
Nevertheless, it is discrete as a topological space.
We say that <math>X</math> 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 [[singleton (mathematics)|singleton]]s are open, which is the case if and only if it does not contain any [[accumulation point]]s.
* The singletons form a [[basis (topology)|basis]] for the discrete topology.
* A uniform space <math>X</math> is discrete if and only if the diagonal <math>\{(x,x) : x \in X\}</math> is an [[entourage (topology)|entourage]].
* Every discrete topological space satisfies each of the [[separation axioms]]; in particular, every discrete space is [[Hausdorff space|Hausdorff]], that is, separated.
* A discrete space is [[compact space|compact]] [[if and only if]] it is [[finite set|finite]].
* Every discrete uniform or metric space is [[complete space|complete]].
* Combining the above two facts, every discrete uniform or metric space is [[totally bounded space|totally bounded]] if and only if it is finite.
* Every discrete metric space is [[bounded space|bounded]].
* Every discrete space is [[first-countable space|first-countable]]; it is moreover [[second-countable space|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 <math>X</math> is a topological space and <math>Y</math> is a set carrying the discrete topology, then <math>X</math> is evenly covered by <math>X \times Y</math> (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 <math>\mathbb{R}</math> (with its usual [[Euclidean topology]]) that is discrete is necessarily [[Countable set|countable]].{{sfn | Wilansky | 2008 | p=35}}

Any function from a discrete topological space to another topological space is [[continuous function (topology)|continuous]], and any function from a discrete uniform space to another uniform space is [[uniformly continuous]]. That is, the discrete space <math>X</math> is [[free object|free]] on the set <math>X</math> in the [[category theory|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 [[morphism]]s. 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 [[Mathematical structure|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 map]]s; 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 space]]s 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 <math>f</math> from a topological space <math>Y</math> to a discrete space <math>X</math> is continuous if and only if it is ''[[locally constant function|locally constant]]'' in the sense that every point in <math>Y</math> has a [[topological neighborhood|neighborhood]] on which <math>f</math> is constant.

Every [[Ultrafilter (set theory)|ultrafilter]] <math>\mathcal{U}</math> on a non-empty set <math>X</math> can be associated with a topology <math>\tau = \mathcal{U} \cup \left\{ \varnothing \right\}</math> on <math>X</math> with the property that {{em|every}} non-empty proper subset <math>S</math> of <math>X</math> is {{em|either}} an [[Open set|open subset]] or else a [[Closed set|closed subset]], but never both. Said differently, {{em|every}} subset is open [[Logical disjunction|or]] closed but (in contrast to the discrete topology) the {{em|only}} subsets that are {{em|both}} open and closed (i.e. [[clopen]]) are <math>\varnothing</math> and <math>X</math>. In comparison, {{em|every}} subset of <math>X</math> is open [[Logical conjunction|and]] closed in the discrete topology.

==Examples and 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 (mathematics)|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 group]]s". 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 and countable topological space (an uncountable discrete space is not second-countable). We can therefore view any discrete countable group as a 0-dimensional [[Lie group]].

A [[Product topology|product]] of [[countably infinite]] copies of the discrete space of [[natural number]]s is [[homeomorphic]] to the space of [[irrational number]]s, with the homeomorphism given by the [[simple continued fraction|continued fraction expansion]]. A product of countably infinite copies of the discrete space [[2 (number)|<math>\{0,1\}</math>]] 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 numeral system|ternary notation]] of numbers. (See [[Cantor space]].) Every [[Fiber (mathematics)|fiber]] of a [[locally injective function]] is necessarily a discrete subspace of its [[Domain of a function|domain]]. 

In the [[foundations of mathematics]], the study of [[Compact space|compactness]] properties of products of <math>\{0,1\}</math> is central to the topological approach to the [[ultrafilter lemma]] (equivalently, the [[Boolean prime ideal theorem]]), which is a weak form of the [[axiom of choice]].

== Indiscrete spaces ==
{{main|Trivial topology}}

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.

== See also ==

* [[Cylinder set]]
* [[List of topologies]]
* [[Taxicab geometry]]

== References ==
{{reflist}}
<!--See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for an explanation of how to generate footnotes using the <ref(erences/)> tags-->
* {{cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | edition=2nd | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=3-540-90312-7 | mr=507446 | zbl=0386.54001 }}
* {{Wilansky Topology for Analysis 2008}} <!-- {{sfn | Wilansky | 2008 | p=}} -->

{{Metric spaces}}

[[Category:General topology]]
[[Category:Metric spaces]]
[[Category:Topological spaces]]
[[Category:Topology]]