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:

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

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

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Distinct points are separated.
Title  T1 space 
Canonical name  T1Space 
Date of creation  20130322 12:18:14 
Last modified on  20130322 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 