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Concept
Discrete space
Summary
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 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 in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.
Given a set :
A metric space is said to be uniformly discrete if there exists a 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
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 (with metric inherited from the real line 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 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 doesn't contain any accumulation points.
The singletons form a basis 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 separation axioms; in particular, every discrete space is Hausdorff, that is, separated.
A discrete space is compact if and only if it is finite.
Every discrete uniform or metric space is complete.
Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it is finite.
Official source
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