Editing Discrete space (section)

== 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 <math>X = \{n^{-1} : n \in \N\}</math> (with metric inherited from the [[real line]] and given by <math>d(x,y) = \left|x - y\right|</math>).
This is not the discrete metric; also, this space is not [[complete (topology)|complete]] and hence not discrete as a uniform space.
Nevertheless, it is discrete as a topological space.
We say that <math>X</math> is ''topologically discrete'' but not ''uniformly discrete'' or ''metrically discrete''.

Additionally:
* The [[topological dimension]] of a discrete space is equal to 0.
* A topological space is discrete if and only if its [[singleton (mathematics)|singleton]]s are open, which is the case if and only if it does not contain any [[accumulation point]]s.
* The singletons form a [[basis (topology)|basis]] for the discrete topology.
* A uniform space <math>X</math> is discrete if and only if the diagonal <math>\{(x,x) : x \in X\}</math> is an [[entourage (topology)|entourage]].
* Every discrete topological space satisfies each of the [[separation axioms]]; in particular, every discrete space is [[Hausdorff space|Hausdorff]], that is, separated.
* A discrete space is [[compact space|compact]] [[if and only if]] it is [[finite set|finite]].
* Every discrete uniform or metric space is [[complete space|complete]].
* Combining the above two facts, every discrete uniform or metric space is [[totally bounded space|totally bounded]] if and only if it is finite.
* Every discrete metric space is [[bounded space|bounded]].
* Every discrete space is [[first-countable space|first-countable]]; it is moreover [[second-countable space|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]] are [[homeomorphic]].
* Every discrete space is metrizable (by the discrete metric).
* A finite space is metrizable only if it is discrete.
* If <math>X</math> is a topological space and <math>Y</math> is a set carrying the discrete topology, then <math>X</math> is evenly covered by <math>X \times Y</math> (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 <math>\mathbb{R}</math> (with its usual [[Euclidean topology]]) that is discrete is necessarily [[Countable set|countable]].{{sfn | Wilansky | 2008 | p=35}}

Any function from a discrete topological space to another topological space is [[continuous function (topology)|continuous]], and any function from a discrete uniform space to another uniform space is [[uniformly continuous]]. That is, the discrete space <math>X</math> is [[free object|free]] on the set <math>X</math> in the [[category theory|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 [[Mathematical structure|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 <math>f</math> from a topological space <math>Y</math> to a discrete space <math>X</math> is continuous if and only if it is ''[[locally constant function|locally constant]]'' in the sense that every point in <math>Y</math> has a [[topological neighborhood|neighborhood]] on which <math>f</math> is constant.

Every [[Ultrafilter (set theory)|ultrafilter]] <math>\mathcal{U}</math> on a non-empty set <math>X</math> can be associated with a topology <math>\tau = \mathcal{U} \cup \left\{ \varnothing \right\}</math> on <math>X</math> with the property that {{em|every}} non-empty proper subset <math>S</math> of <math>X</math> is {{em|either}} an [[Open set|open subset]] or else a [[Closed set|closed subset]], but never both. Said differently, {{em|every}} subset is open [[Logical disjunction|or]] closed but (in contrast to the discrete topology) the {{em|only}} subsets that are {{em|both}} open and closed (i.e. [[clopen]]) are <math>\varnothing</math> and <math>X</math>. In comparison, {{em|every}} subset of <math>X</math> is open [[Logical conjunction|and]] closed in the discrete topology.