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 $\epsilon >0$ and all $x\in X$ there is $y\in D$ such that .

For example, both the rationals $\mathbb{Q}$ and the irrationals $ℝ\setminus \mathbb{Q}$ 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 ${\mathrm{\aleph }}_0$ if it would otherwise be finite; with this convention, the spaces of density ${\mathrm{\aleph }}_0$ 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 $|X|\le 2^{2^{d(X)}}$.

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