T1 space

A topological space $(X,\tau )$ is said to be $T_1$ (or said to hold the $T_1$ axiom) if for all distinct points $x,y\in X$ ($x\ne y$), there exists an open set $U\in \tau$ such that $x\in U$ and $y\notin U$.

A space being $T_1$ is equivalent to the following statements:

• For every $x\in X$, the set $\{x\}$ is closed.

• Every subset of $X$ is equal to the intersection of all the open sets that contain it.

• Distinct points are separated.

Title	T1 space
Canonical name	T1Space
Date of creation	2013-03-22 12:18:14
Last modified on	2013-03-22 12:18:14
Owner	drini (3)
Last modified by	drini (3)
Numerical id	10
Author	drini (3)
Entry type	Definition
Classification	msc 54D10
Synonym	T1
Related topic	T0Space
Related topic	T2Space
Related topic	T3Space
Related topic	RegularSpace
Related topic	ASpaceIsT1IfAndOnlyIfEverySubsetAIsTheIntersectionOfAllOpenSetsContainingA
Related topic	SierpinskiSpace
Related topic	PropertyThatCompactSetsInASpaceAreClosedLiesStrictlyBetweenT1AndT2