In topology and related branches of mathematics, a T_{1} space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R_{0} space is one in which this holds for every pair of topologically distinguishable points. The properties T_{1} and R_{0} are examples of separation axioms.
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Definitions
Let X be a topological space and let x and y be points in X. We say that x and y can be separated if each lies in an open set which does not contain the other point.
A T_{1} space is also called an accessible space or a Fréchet space and a R_{0} space is also called a symmetric space. (The term Fréchet space also has an entirely different meaning in functional analysis. For this reason, the term T_{1} space is preferred. There is also a notion of a FréchetUrysohn space as a type of sequential space. The term symmetric space has another meaning.)
Properties
Let X be a topological space. Then the following conditions are equivalent:
 X is a T_{1} space.
 X is a T_{0} space and a R_{0} space.
 Points are closed in X; i.e. given any x in X, the singleton set {x} is a closed set.
 Every subset of X is the intersection of all the open sets containing it.
 Every finite set is closed.
 Every cofinite set of X is open.
 The fixed ultrafilter at x converges only to x.
 For every point x in X and every subset S of X, x is a limit point of S if and only if every open neighbourhood of x contains infinitely many points of S.
Let X be a topological space. Then the following conditions are equivalent:
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