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 space is 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 |