Definitions
Given a set X:
- the discrete topology on X is defined by letting every subset of X be open (and hence also closed), and X is a discrete topological space if it is equipped with its discrete topology;
- the discrete uniformity on X is defined by letting every superset of the diagonal {(x,x) : x is in X} in X × X be an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.
- the discrete metric on X is defined by
for any . In this case is called a discrete metric space or a space of isolated points.
- a set S is discrete in a metric space, for, if for every, there exists some (depending on ) such that for all ; such a set consists of isolated points. A set S is uniformly discrete in the metric space, for, if there exists ε > 0 such that for any two distinct, > ε.
A metric space is said to be uniformly discrete if there exists 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 {1, 1/2, 1/4, 1/8, ...} of real numbers.
Read more about this topic: Discrete Space
Famous quotes containing the word definitions:
“The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
—Pamela Patrick Novotny (20th century)
“Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
—Edmond De Goncourt (1822–1896)