Alexandroff space

Topological spaceMathworldPlanetmath X is called Alexandroff if the intersectionMathworldPlanetmath 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 denote the set of real numbers and let τ={[a,)|a}{(b,)|b}. Then τ is a topologyMathworldPlanetmath on and (,τ) is an Alexandroff space.

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

Ao={UX|U is open and A is contained in U}.

Then Ao 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