Discrete space: Difference between revisions

Content deleted Content added

Inline

Revision as of 17:44, 3 March 2021

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.

Definitions

Given a set X:

the discrete topology on X is defined by letting every
entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.
the discrete metric

\rho

on X is defined by

\rho (x,y)={\begin{cases}1&{\mbox{if}}\ x\neq y,\\0&{\mbox{if}}\ x=y\end{cases}}

for any

x,y\in X

. In this case

(X,\rho )

is called a discrete metric space or a space of isolated points.

a set S is discrete in a metric space

(X,d)

, for

S\subseteq X

, if for every

x\in S

, there exists some

\delta >0

(depending on

x

) such that

d(x,y)>\delta

for all

y\in S\setminus \{x\}

; such a set consists of isolated points. A set S is uniformly discrete in the metric space

(X,d)

, for

S\subseteq X

, if there exists ε > 0 such that for any two distinct

x,y\in S

,

d(x,y)>\varepsilon

.

A metric space $(E,d)$ is said to be

uniformly discrete

if there exists a "packing radius"

r>0

such that, for any

x,y\in E

, one has either

x=y

or

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 {1, 1/2, 1/4, 1/8, ...} of real numbers.

Proof that a discrete space is not necessarily uniformly discrete

Let X = {1, 1/2, 1/4, 1/8, ...}, consider this set using the usual metric on the real numbers. Then, X is a discrete space, since for each point 1/2ⁿ, we can surround it with the interval (1/2ⁿ − ɛ, 1/2ⁿ + ɛ), where ɛ = 1/2(1/2ⁿ − 1/2ⁿ⁺¹) = 1/2ⁿ⁺². The intersection (1/2ⁿ − ɛ, 1/2ⁿ + ɛ) ∩ {1/2ⁿ} is just the singleton {1/2ⁿ}. Since the intersection of two open sets is open, and singletons are open, it follows that X is a discrete space.

However, X cannot be uniformly discrete. To see why, suppose there exists an r > 0 such that d(x, y) > r whenever x ≠ y. It suffices to show that there are at least two points x and y in X that are closer to each other than r. Since the distance between adjacent points 1/2ⁿ and 1/2ⁿ⁺¹ is 1/2ⁿ⁺¹, we need to find an n that satisfies this inequality:

{\begin{aligned}{\frac {1}{2^{n+1}}}&<r\\1&<r2^{n+1}\\{\frac {1}{r}}&<2^{n+1}\\\log _{2}\left({\frac {1}{r}}\right)&<n+1\\-\log _{2}(r)&<n+1\\-1-\log _{2}(r)&<n\end{aligned}}

Since there is always an n bigger than any given real number, it follows that there will always be at least two points in X that are closer to each other than any positive r, therefore X 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 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
countable
.
Every discrete space is
totally disconnected
.
Every non-empty discrete space is
second category
.
Any two discrete spaces with the same
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
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

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

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

neighborhood

on which f is constant.
Every ultrafilter ${\mathcal {U}}$ on a non-empty set $X$ can be associated with a topology $\tau :={\mathcal {U}}\cup \{\varnothing \}$ on $X$ with the property that every non-empty proper subset $S$ of $X$ is either an
clopen
) are $\varnothing$ and $X.$ In comparison, every subset of $X$ is open and closed in the discrete topology.

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. 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
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.

Indiscrete spaces

Main article: Trivial topology

In some ways, the opposite of the discrete topology is the
cofree
: every function from a topological space to an indiscrete space is continuous, etc.

See also

Cylinder set

List of topologies

Taxicab geometry

References

Zbl 0982.52018
.

^ Wilansky 2008, p. 35.

Zbl 0386.54001
.

ISBN 1-57955-008-8
.

OCLC 227923899
.

[1] Zbl 0982.52018
.

[FOOTNOTEWilansky200835-2] Wilansky 2008, p. 35.

[1]

[2]

@@ Line 99: / Line 99: @@
 <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 }}
+* {{cite book|last=Wolfram|first=Stephen|authorlink=Stephen Wolfram|title=A New Kind of Science|url=https://www.wolframscience.com/nks/|publisher=Wolfram Media, Inc.|year=2002|page=[https://www.wolframscience.com/nks/notes-9-6--history-of-discrete-models-of-space/ 1027]|isbn=1-57955-008-8}}
 * {{Wilansky Topology for Analysis 2008}} <!-- {{sfn | Wilansky | 2008 | p=}} -->