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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 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, i.e., it defines all subsets as open sets. In particular, each [[Singleton (mathematics)|singleton]] is an open set in the discrete topology.
 
== Definitions ==
Given a set ''X'':
* the '''discrete topology''' on ''X'' is defined by letting every [[subset]] of ''X'' be [[open set|open]] (and hence also [[closed set|closed]]), and ''X'' is a '''discrete topological space''' if it is equipped with its discrete topology;
* the '''discrete [[uniform space|uniformity]]''' on ''X'' is defined by letting every [[superset]] of the diagonal {(''x'',''x'')&nbsp;: ''x'' is in ''X''} in ''X''&nbsp;× ''X'' be an [[Entourage (topology)#Entourage definition|entourage]], and ''X'' is a '''discrete uniform space''' if it is equipped with its discrete uniformity.
* the '''discrete [[metric space|metric]]''' <math>\rho</math> on ''X'' is defined by
:<math>\rho(x,y) =
\left\{\begin{matrix}
1 &\mbox{if}\ x\neq y , \\
0 &\mbox{if}\ x = y
\end{matrix}\right.
</math>
for any <math>x,y \in X</math>. In this case <math>(X,\rho)</math> is called a '''discrete metric space''' or a '''space of [[isolated point]]s'''.
* a [[set (mathematics)|set]] ''S'' 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 ''S'' is '''uniformly discrete''' in the [[metric space]] <math>(X,d)</math>, for <math>S \subseteq X</math>, if there exists ''ε'' > 0 such that for any two distinct <math>x, y \in S</math>, <math>d(x, y)</math> >  ''ε''.
 
A metric space <math>(E,d)</math> is said to be ''[[uniformly discrete set|uniformly discrete]]'' if there exists a "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 {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''&nbsp;:= {1/''n''&nbsp;: ''n''&nbsp;= 1,2,3,...} (with metric inherited from the [[real line]] and given by d(''x'',''y'')&nbsp;= |''x''&nbsp;&minus; ''y''|).
Obviously, 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 ''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 [[singleton (mathematics)|singleton]]s are open, which is the case if and only if it doesn't contain any [[accumulation point]]s.
* The singletons form a [[basis (topology)|basis]] for the discrete topology.
* A uniform space ''X'' is discrete if and only if the diagonal {(''x'',''x'')&nbsp;: ''x'' is in ''X''} 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 with at least two points 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 {{nowrap|''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 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 ''X'' is [[free object|free]] on the set ''X'' 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 ''f'' from a topological space ''Y'' to a discrete space ''X'' is continuous if and only if it is ''[[locally constant function|locally constant]]'' in the sense that every point in ''Y'' has a [[topological neighborhood|neighborhood]] on which ''f'' is constant.
 
== 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 analytical manifold) is nothing but a discrete topological space. We can therefore view any discrete 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 [[continued fraction]] expansion. A product of countably infinite copies of the discrete space [[2 (number)|{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 numeral system|ternary notation]] of numbers. (See [[Cantor space]].)
 
In the [[foundations of mathematics]], the study of [[compact space|compactness]] properties of products of {0,1} is central to the topological approach to the [[Boolean prime ideal theorem|ultrafilter principle]], which is a weak form of [[axiom of choice|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.
 
== Quotation ==
*[[Stanislaw Ulam]] characterized [[Los Angeles, California|Los Angeles]] as "a discrete space, in which there is an hour's drive between points".<ref>[[Stanislaw Ulam]]'s autobiography, ''Adventures of a Mathematician''.</ref>
 
==See also==
* [[Cylinder set]]
 
== References ==
<!--See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for an explanation of how to generate footnotes using the <ref(erences/)> tags-->
<references />
* {{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 }}
 
[[Category:Topology]]
[[Category:General topology]]
[[Category:Topological spaces]]

Latest revision as of 10:27, 4 October 2014

If you are the proud owner of a tractor, the day will come when you will be available in the market for tires to suit your machine. To save money, many people look for a very good deal on used tractor tires It would be a good idea so that you can do your homework, and learn as much as possible about these tires earlier than you got down to find the proper tire , or tires that have been pre-owned.

With a view to keep the consistent excellence, the experts of Nitto tires incorporated high-level of laptop-modeled tread designs in order to reduce pattern noise of the tires at the time of touring. Together with this, polymer treads compounds additionally helps to current high-finish efficiency to the customers while touring on roads. It also enhances the tread life of the tires making the journey probably the most memorable one in your complete life. Additional highway-biting traction on each moist and dry roads, also makes it an all season tire for its clients. These underlining mind-blowing specifications improves the variety of Nitto Tires For Sale in the market.

It may be fairly difficult to get shot of previous, used tires In the course of the past, tires had been merely burned as a manner of disposing of them. Everybody knows nonetheless that this can be most likely hazardous for people and the environment. Some folks also used to stack used tires in a nook of their yards or garages. This is additionally not a fantastic concept although. Old tires can merely collect water where mosquitoes can lay their eggs on. Off road tires are completely different from the traditional tires as a result of these tires usually are not meant for normal asphalt roads. Off the street automobiles embody wheel loaders, backhoes, graders, trenchers, in addition to giant mining vans, with four-wheel drives. read extra The way to Put Fluid in Tires How you can Compare 4WD Tires Find out how to Lace Tires for Delivery Tire Plies

The tire is built over a period of time that normally takes over ten minutes (but in big, automized factories can take as little as three minutes!) and for much longer for big tires and racing tires (which are made utilizing particular machinary and a whole lot of guide work). The method begins with mixing the new rubber with carbon black (an inexpensive anti-radiant and heat-ressistant substance) and the other chemical compounds talked about above, which are than cut into long "mats" of 80 centimeters. The mats are than minimize and extruded to create a single tread, sidewall or innerliner.

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