# boundary / frontier

Definition.
Let $X$ be a topological space^{} and let $A$ be a subset
of $X$. The *boundary* (or *frontier*) of $A$ is the set
$\partial A=\overline{A}\cap \overline{X\backslash A}$,
where the overline denotes the closure^{} of a set.
Instead of $\partial A$, many authors use some other notation
such as $\mathrm{bd}(A)$, $\mathrm{fr}(A)$, ${A}^{b}$ or $\beta (A)$.
Note that the $\partial $ symbol is also used for other meanings of ‘boundary’.

From the definition, it follows that the boundary of any set is a closed set. It also follows that $\partial A=\partial (X\backslash A)$, and $\partial X=\mathrm{\varnothing}=\partial \mathrm{\varnothing}$.

The term ‘boundary’ (but not ‘frontier’) is used in a different sense for topological manifolds^{}: the boundary $\partial M$ of a topological $n$-manifold $M$ is the set of points in $M$ that do not have a neighbourhood homeomorphic to ${\mathbb{R}}^{n}$. (Some authors define topological manifolds in such a way that they necessarily have empty boundary.)
For example, the boundary of the topological $1$-manifold $[0,1]$ is $\{0,1\}$.

Title | boundary / frontier |
---|---|

Canonical name | BoundaryFrontier |

Date of creation | 2013-03-22 13:34:46 |

Last modified on | 2013-03-22 13:34:46 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 17 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54-00 |

Synonym | boundary |

Synonym | frontier |

Synonym | topological boundary |

Related topic | ExtendedBoundary |

Related topic | Interior |