Sierpinski space

Sierpinski space is the topological spaceMathworldPlanetmath X={x,y} with the topology given by {X,{x},}.

Sierpinski space is T0 ( but not T1 ( It is T0 because {x} is the open set containing x but not y. It is not T1 because every open set U containing y (namely X) contains x (in other words, there is no open set containing y but not containing x).

Remark. From the Sierpinski space, one can construct many non-T1 T0 spaces, simply by taking any set X with at least two elements, and take any non-empty proper subsetMathworldPlanetmathPlanetmath UX, and set the topology 𝒯 on X by 𝒯=P(U){X}.

Title Sierpinski space
Canonical name SierpinskiSpace
Date of creation 2013-03-22 12:06:26
Last modified on 2013-03-22 12:06:26
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 54G20
Synonym Sierpiński space
Related topic T1Space
Related topic T2Space
Related topic SeparationAxioms