Alexandroff space is T1 if and only if it is discrete


PropositionPlanetmathPlanetmathPlanetmath. Let X be an Alexandroff space. Then X is T1 if and only if X is discrete.

Proof. ,,” It is easy to see, that every discrete space is Alexandroff and T1.

,,” Recall that topological spaceMathworldPlanetmath is T1 if and only if every subset is equal to the intersectionMathworldPlanetmath of all its open neighbourhoods. So let xX. Then the intersection of all open neighbourhoods {x}o of x is equal to {x}. But since X is Alexandroff, then {x}o={x} is open and thus points are open. Therefore X is discrete.

Title Alexandroff space is T1 if and only if it is discrete
Canonical name AlexandroffSpaceIsT1IfAndOnlyIfItIsDiscrete
Date of creation 2013-03-22 18:46:08
Last modified on 2013-03-22 18:46:08
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Derivation
Classification msc 54A05