Discrete Space - Uses

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 analytical manifold) is nothing but a discrete topological space. We can therefore view any discrete group as a 0-dimensional Lie group.

A product of countably infinite copies of the discrete space of natural numbers is homeomorphic to the space of irrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countably infinite copies of the discrete space {0,1} 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 ternary notation of numbers. (See Cantor space.)

In the foundations of mathematics, the study of compactness properties of products of {0,1} is central to the topological approach to the ultrafilter principle, which is a weak form of choice.