# closure

The *closure ^{}* $\overline{A}$ of a subset $A$ of a topological space

^{}$X$ is the intersection

^{}of all closed sets

^{}containing $A$.

Equivalently, $\overline{A}$ consists of $A$ together with all limit points^{} of $A$ in $X$ or equivalently $x\in \overline{A}$ if and only if every neighborhood^{} of $x$ intersects $A$. Sometimes the notation $\mathrm{cl}(A)$ is used.

If it is not clear, which topological space is used, one writes ${\overline{A}}^{X}$. Note that if $Y$ is a subspace^{} of $X$, then ${\overline{A}}^{X}$ may not be the same as ${\overline{A}}^{Y}$. For example, if $X=\mathbb{R}$, $Y=(0,1)$ and $A=(0,1)$, then ${\overline{A}}^{X}=[0,1]$ while ${\overline{A}}^{Y}=(0,1)$.

Title | closure |
---|---|

Canonical name | Closure |

Date of creation | 2013-03-22 12:05:40 |

Last modified on | 2013-03-22 12:05:40 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 9 |

Author | mathwizard (128) |

Entry type | Definition |

Classification | msc 54A99 |

Related topic | ClosureAxioms |

Related topic | Interior |