dense set
A subset of a topological space is said to be dense (or everywhere dense) in if the closure
of is equal to .Equivalently, is dense if and only if intersects every nonempty open set.
In the special case that is a metric space with metric ,then this can be rephrased as:for all and all there is such that .
For example, both the rationals and the irrationals are dense in the reals .
The least cardinality of a dense set of a topological spaceis called the density of the space.It is conventional to take the density to be if it would otherwise be finite;with this convention,the spaces of density are precisely the separable spaces.The density of a topological space is denoted .If is a Hausdorff space,it can be shown that .
Title | dense set |
Canonical name | DenseSet |
Date of creation | 2013-03-22 12:05:42 |
Last modified on | 2013-03-22 12:05:42 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 12 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 54A99 |
Synonym | dense subset |
Synonym | everywhere dense set |
Synonym | everywhere dense subset |
Synonym | everywhere-dense set |
Synonym | everywhere-dense subset |
Related topic | NowhereDense |
Related topic | DenseInAPoset |
Defines | dense |
Defines | everywhere dense |
Defines | everywhere-dense |
Defines | density |