Editing Discrete space (section)

==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 (mathematics)|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 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]].

A [[Product topology|product]] 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 [[simple continued fraction|continued fraction expansion]]. A product of countably infinite copies of the discrete space [[2 (number)|<math>\{0,1\}</math>]] 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 numeral system|ternary notation]] of numbers. (See [[Cantor space]].) Every [[Fiber (mathematics)|fiber]] of a [[locally injective function]] is necessarily a discrete subspace of its [[Domain of a function|domain]]. 

In the [[foundations of mathematics]], the study of [[Compact space|compactness]] properties of products of <math>\{0,1\}</math> is central to the topological approach to the [[ultrafilter lemma]] (equivalently, the [[Boolean prime ideal theorem]]), which is a weak form of the [[axiom of choice]].