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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.

1 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 1, 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".

2 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.
• The singletons form a basis for the discrete topology.
• 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 HausdorffIn topology and related branches of mathematics, Hausdorff spaces and preregular spaces are kinds of topological spaces. The conditions imposed are the most significant separation axioms. Definitions Suppose that X is a topological space. X is a Hausdorff, aka separated.
• A discrete space is compactIn mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R n in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if e iffIn mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if . It is often, not always, written italicized: iff''. Although "P iff Q" is most standard, common alternative phrases include "P it is finiteIn mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2,. n with n isin N . It is a theorem that a set is finite if and only if there exists no bijection between the set and any of its prop.
• 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 non-empty 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. 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 one and nonexpansive 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 one is nonexpansive.

Going the other direction, 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.