T1 space

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In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms.

Contents

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 T1 space is also called an accessible space or a Fréchet space and a R0 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 T1 space is preferred. There is also a notion of a Fréchet-Urysohn 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 T1 space.
  • X is a T0 space and a R0 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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