In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.
Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.
is said to be uniformly discrete if there exists a packing radius
consider this set using the usual metric on the real numbers.
Since the intersection of an open set of the real numbers and
bigger than any given real number, it follows that there will always be at least two points in
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
(with metric inherited from the real line and given by
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 1 and short maps.
is continuous if and only if it is locally constant in the sense that every point in
with the property that every non-empty proper subset
Said differently, every subset is open or closed but (in contrast to the discrete topology) the only subsets that are both open and closed (i.e. clopen) are
is open and closed in the discrete topology.
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.
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).
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
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 notation of numbers.
Every fiber of a locally injective function is necessarily a discrete subspace of its domain.
In the foundations of mathematics, the study of compactness properties of products of
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.
In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), which has the fewest possible open sets (just the empty set and the space itself).