In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.
Definitions
Given a set X:
- the discrete topology on X is defined by letting every subset of X be open, and X is a discrete topological space if it is equipped with its discrete topology;
- the discrete uniformity on X is defined by letting every superset of the diagonal {(x,x) : x ∈ X} in X × X be an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.
- the discrete metric on X is defined by letting the distance between any distinct points x and y be one, and X is a discrete metric space if it is equipped with its discrete metric.
There are many other types of discrete structures that can be placed on a set, but only these cases (to the knowledge of the authors so far of this article) are generally called "spaces".
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|). Obviously, 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:
- 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.
- A uniform space X is discrete if and only if the diagonal {(x,x) : x ∈ X} is an entourage.
- Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space is Hausdorff, aka separated.
- A discrete space is compact iff 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 iff it is finite.
- Every discrete metric space is bounded.
- Every discrete space is first countable, and a discrete space is second countable iff it is countable.
- Every discrete space is totally disconnected.
- Every discrete space is second category.
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. However, this is not true of the discrete metric space in the category of metric spaces and contraction mappings. Nevertheless, the discrete metric space is free in the category of bounded metric spaces and contraction mappings; that is, any function from a discrete metric space to a bounded metric space is a contraction mapping. These facts are examples of a much broader phenomenon, in which discrete structures are usually free on sets.
Going the other diretion, a function f from a topological space Y to a discrete space X is continuous if and only it if is locally constant in the sense that every point in Y has a neighborhood on which f is constant.
While discrete spaces are rather uninteresting from a topological viewpoint, one can easily construct interesting spaces from them. For instance, a product of countably infinitely many copies of the discrete space {0,1} is homeomorphic to the Cantor set and a product of countably infinitely many copies of the discrete space of natural numbers is homeomorphic to the space of irrational numbers. (To try to construct these homeomorphisms yourself, think ternary notation in the case of the Cantor set and continued fractions in the case of the irrational numbers.)