In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a, 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
A metric space
(E,d)
r>0
x,y\inE,
x=y
d(x,y)>r.
\left\{2-n:n\in\N0\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
X=\{n-1:n\in\N\}
d(x,y)=\left|x-y\right|
X
Additionally:
X
\{(x,x):x\inX\}
X
Y
X
X x Y
R
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
X
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
Y
X
Y
f
l{U}
X
\tau=l{U}\cup\left\{\varnothing\right\}
X
S
X
\varnothing
X
X
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 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 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.) Every fiber of a locally injective function is necessarily a discrete subspace of its domain.
In the foundations of mathematics, the study of compactness properties of products of
\{0,1\}
See main article: 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.