# 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 $\mathbb{R}\setminus \mathbb{Q}$ are dense in the reals $\mathbb{R}$.

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 |