, 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
is said to be ''uniformly discrete
'' if there exists a "packing radius"
such that, for any
, one has either
The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set of real numbers.
Let , consider this set using the usual metric on the real numbers. Then, ''X'' is a discrete space, since for each point 1/2''n''
, we can surround it with the interval , where . The intersection is just the singleton . 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 such that whenever . 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 1/2''n''
, we need to find an ''n'' that satisfies this inequality:
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.
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'' := (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''.
* The topological dimension
of a discrete space is equal to 0.
* A topological space is discrete if and only if its singleton
s are open, which is the case if and only if it doesn't contain any accumulation point
* The singletons form a basis
for the discrete topology.
* A uniform space ''X'' is discrete if and only if the diagonal 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 first-countable
; it is moreover 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
* 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 (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 (with its usual Euclidean topology
) that is discrete is necessarily countable
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'' 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 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 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
'' in the sense that every point in ''Y'' has a neighborhood
on which ''f'' is constant.
on a non-empty set
can be associated with a topology
with the property that non-empty proper subset
is an open subset
or else a closed subset
, but never both. Said differently, subset is open or
closed but (in contrast to the discrete topology) the subsets that are open and closed (i.e. clopen
In comparison, subset of
is open and
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, 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 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 topological space. We can therefore view any discrete group as a 0-dimensional Lie group
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
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
In the foundations of mathematics
, the study of compactness
properties of products of is central to the topological approach to the ultrafilter principle
, which is a weak form of choice
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.
* Cylinder set
* List of topologies
* Taxicab geometry