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 .
|Date of creation||2013-03-22 12:05:42|
|Last modified on||2013-03-22 12:05:42|
|Last modified by||yark (2760)|
|Synonym||everywhere dense set|
|Synonym||everywhere dense subset|