T0 space

A topological space $(X,\tau )$ is said to be $T_0$ (or to satisfy the $T_0$ axiom ) if for all distinct $x,y\in X$ there exists an open set $U\in \tau$ such that either $x\in U$ and $y\notin U$ or $x\notin U$ and $y\in U$.

All $T_1$ spaces (http://planetmath.org/T1Space) are $T_0$. An example of $T_0$ space that is not $T_1$ is the $2$-point Sierpinski space.

Title	T0 space
Canonical name	T0Space
Date of creation	2013-03-22 12:18:12
Last modified on	2013-03-22 12:18:12
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	13
Author	yark (2760)
Entry type	Definition
Classification	msc 54D10
Synonym	Kolmogorov space
Synonym	Kolmogoroff space
Related topic	Ball
Related topic	T1Space
Related topic	T2Space
Related topic	RegularSpace
Related topic	T3Space
Defines	T0