Discrete space

 ( General Topology ) 

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are Isolated point from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is Open set in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.

---

Definitions Given a set $X$:

A metric space $(E,d)$ is said to be uniformly discrete if there exists a $r\; >\; 0$ such that, for any $x,y\; \backslash in\; E,$ one has either $x\; =\; y$ or $d(x,y)\; >\; r.$

The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set $\backslash left\backslash \{2^\{-n\}\; :\; n\; \backslash in\; \backslash N\_0\backslash right\backslash \}.$

---

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 $X\; =\; \backslash \{n^\{-1\}\; :\; n\; \backslash in\; \backslash N\backslash \}$ (with metric inherited from the real line and given by $d(x,y)\; =\; \backslash left|x\; -\; y\backslash right|$). 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 $X$ 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 singletons are open, which is the case if and only if it does not contain any accumulation points.
• The singletons form a basis for the discrete topology.
• A uniform space $X$ is discrete if and only if the diagonal $\backslash \{(x,x)\; :\; x\; \backslash in\; X\backslash \}$ is an entourage.
• Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space is Hausdorff space, that is, separated.
• A discrete space is compact space if and only if it is finite set.
• Every discrete uniform or metric space is complete space.
• Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it is finite.
• Every discrete metric space is bounded space.
• Every discrete space is first-countable; it is moreover 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 $X$ is a topological space and $Y$ is a set carrying the discrete topology, then $X$ is evenly covered by $X\; \backslash times\; Y$ (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 $\backslash mathbb\{R\}$ (with its usual Euclidean topology) that is discrete is necessarily Countable set.

Any function from a discrete topological space to another topological space is continuous, and any function from a discrete uniform space to another uniform space is uniformly continuous. That is, the discrete space $X$ is free object on the set $X$ in the category theory 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 . 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 ; 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 $f$ from a topological space $Y$ to a discrete space $X$ is continuous if and only if it is locally constant in the sense that every point in $Y$ has a neighborhood on which $f$ is constant.

Every ultrafilter $\backslash mathcal\{U\}$ on a non-empty set $X$ can be associated with a topology $\backslash tau\; =\; \backslash mathcal\{U\}\; \backslash cup\; \backslash left\backslash \{\; \backslash varnothing\; \backslash right\backslash \}$ on $X$ with the property that non-empty proper subset $S$ of $X$ is an Open set or else a Closed set, but never both. Said differently, subset is open or closed but (in contrast to the discrete topology) the subsets that are open and closed (i.e. clopen) are $\backslash varnothing$ and $X$. In comparison, subset of $X$ is open and closed in the discrete topology.

---

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 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 "". 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 of countably infinite copies of the discrete space of 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 $\backslash \{0,1\backslash \}$ 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. (See Cantor space.) Every fiber of a locally injective function is necessarily a discrete subspace of its domain.

In the foundations of mathematics, the study of Compact space properties of products of $\backslash \{0,1\backslash \}$ 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.

---

Indiscrete spaces 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). 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

• (1978). 9783540903123, Springer-Verlag. ISBN 9783540903123

Post Comment