dense set
A subset D of a topological space X
is said to be dense (or everywhere dense) in X
if the closure
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 x∈X there is y∈D such that d(x,y)<ε.
For example, both the rationals ℚ and the irrationals ℝ∖ℚ are dense in the reals ℝ.
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 ℵ0
if it would otherwise be finite;
with this convention,
the spaces of density ℵ0 are precisely the separable spaces.
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 |