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 :
the discrete topology on is defined by letting every
subset of be
open (and hence also
closed), and is a discrete topological space if it is equipped with its discrete topology;
the discrete uniformity on is defined by letting every
superset of the diagonal in be an
entourage, and is a discrete uniform space if it is equipped with its discrete uniformity.
for any In this case is called a discrete metric space or a space of
isolated points.
a discrete subspace of some given topological space refers to a
topological subspace of (a subset of together with the
subspace topology that induces on it) whose topology is equal to the discrete topology. For example, if has its usual
Euclidean topology then (endowed with the subspace topology) is a discrete subspace of but is not.
a
set is discrete in a
metric space for if for every there exists some (depending on ) such that for all ; such a set consists of
isolated points. A set is uniformly discrete in the
metric space for if there exists such that for any two distinct
A metric space is said to be uniformly discrete if there exists a packing radius such that, for any one has either or [1] The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set
Proof that a discrete space is not necessarily uniformly discrete
Let consider this set using the usual metric on the real numbers. Then, is a discrete space, since for each point we can surround it with the open interval where The intersection is therefore trivially the singleton Since the intersection of an open set of the real numbers and is open for the induced topology, it follows that is open so singletons are open and is a discrete space.
However, 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 and in that are closer to each other than Since the distance between adjacent points and is we need to find an that satisfies this inequality:
Since there is always an bigger than any given real number, it follows that there will always be at least two points in that are closer to each other than any positive therefore 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 (with metric inherited from the
real line and given by ).
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 is topologically discrete but not uniformly discrete or metrically discrete.
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 is
free on the set 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
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 from a topological space to a discrete space is continuous if and only if it is locally constant in the sense that every point in has a
neighborhood on which is constant.
Every
ultrafilter on a non-empty set can be associated with a topology on with the property that every non-empty proper subset of is either an
open subset or else a
closed subset, but never both. Said differently, every subset is open
or closed but (in contrast to the discrete topology) the only subsets that are both open and closed (i.e.
clopen) are and . In comparison, every subset of is open
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 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.
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.
^Pleasants, Peter A.B. (2000). "Designer quasicrystals: Cut-and-project sets with pre-assigned properties". In Baake, Michael (ed.). Directions in mathematical quasicrystals. CRM Monograph Series. Vol. 13. Providence, RI:
American Mathematical Society. pp. 95–141.
ISBN0-8218-2629-8.
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