# Alexandroff space

Topological space^{} $X$ is called Alexandroff if the intersection^{} of every family of open sets is open.

Of course every finite topological space is Alexandroff, but there are also bigger Alexandroff spaces. For example let $\mathbb{R}$ denote the set of real numbers and let $\tau =\{[a,\mathrm{\infty})|a\in \mathbb{R}\}\cup \{(b,\mathrm{\infty})|b\in \mathbb{R}\}$. Then $\tau $ is a topology^{} on $\mathbb{R}$ and $(\mathbb{R},\tau )$ is an Alexandroff space.

If $X$ is an Alexandroff space and $A\subseteq X$, then we may talk about smallest open neighbourhood of $A$. Indeed, let

$${A}^{o}=\bigcap \{U\subseteq X|U\text{is open and}A\text{is contained in}U\}.$$ |

Then ${A}^{o}$ is open.

Title | Alexandroff space |
---|---|

Canonical name | AlexandroffSpace |

Date of creation | 2013-03-22 18:45:41 |

Last modified on | 2013-03-22 18:45:41 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 5 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 54A05 |