dense set

A subset D of a topological spaceMathworldPlanetmath X is said to be dense (or everywhere dense) in X if the closureMathworldPlanetmathPlanetmath of D is equal to X. Equivalently, D is dense if and only if D intersects every nonempty open set.

In the special case that X is a metric space with metric d, then this can be rephrased as: for all ε>0 and all xX there is yD such that d(x,y)<ε.

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 0 if it would otherwise be finite; with this convention, the spaces of density 0 are precisely the separable spacesMathworldPlanetmath. The density of a topological space X is denoted d(X). If X is a Hausdorff space, it can be shown that |X|22d(X).

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