DefinitionsGiven a set : A metric space is said to be '' uniformly discrete'' if there exists a ''packing radius'' such that, for any , one has either or The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set . Let , consider this set using the usual metric on the real numbers. Then, is a discrete space, since for each point , we can surround it with the interval , where . The intersection is therefore trivially the singleton . Since the intersection of two open sets is open, and singletons are open, it follows that is a discrete space. However, cannot be uniformly discrete. To see why, suppose there exists an such that whenever . It suffices to show that there are at least two points and in that are closer to each other than . Since the distance between adjacent points and is , we need to find an that satisfies this inequality: : Since there is always an bigger than any given real number, it follows that there will always be at least two points in that are closer to each other than any positive , therefore is not uniformly discrete.
PropertiesThe 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 (with metric inherited from the and given by ). 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 is ''topologically discrete'' but not ''uniformly discrete'' or ''metrically discrete''. Additionally: * The of a discrete space is equal to 0. * 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 s. * The singletons form a for the discrete topology. * A uniform space is discrete if and only if the diagonal is an entourage. * Every discrete topological space satisfies each of the ; in particular, every discrete space is , that is, separated. * A discrete space is it is . * Every discrete uniform or metric space is complete. * Combining the above two facts, every discrete uniform or metric space is if and only if it is finite. * Every discrete metric space is bounded. * Every discrete space is ; it is moreover if and only if it is . * 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 is a topological space and is a set carrying the discrete topology, then is evenly covered by (the projection map is the desired covering) * The subspace topology on the integers as a subspace of the is the discrete topology. * A discrete space is separable if and only if it is countable. * Any topological subspace of (with its usual Euclidean topology) that is discrete is necessarily Countable set, countable. 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 is free object, free on the set 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 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 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 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. 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 from a topological space to a discrete space is continuous if and only if it is ''locally constant function, locally constant'' in the sense that every point in has a topological neighborhood, neighborhood on which is constant. Every Ultrafilter (set theory), ultrafilter on a non-empty set can be associated with a topology on with the property that non-empty proper subset of is an Open set, open subset or else a Closed set, closed subset, but never both. Said differently, subset is open Logical disjunction, or closed but (in contrast to the discrete topology) the subsets that are open and closed (i.e. clopen) are and . In comparison, subset of is open Logical conjunction, and closed in the discrete topology.
UsesA 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 groups" . 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 topological space. We can therefore view any discrete group as a 0-dimensional Lie group. A product topology, 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 2 (number), 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.) In the foundations of mathematics, the study of compact space, compactness properties of products of is central to the topological approach to the Boolean prime ideal theorem, ultrafilter principle, which is a weak form of axiom of choice, choice.
Indiscrete spacesIn 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). Where the discrete topology is initial or free, the indiscrete topology is final or cofree: every function ''from'' a topological space ''to'' an indiscrete space is continuous, etc.
See also* Cylinder set * List of topologies * Taxicab geometry