T1 space

DISPLAYTITLE:T1 space In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. 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. Let X be a topological space and let x and y be points in X. We say that x and y are if each lies in a neighbourhood that does not contain the other point. X is called a T1 space if any two distinct points in X are separated. X is called an R0 space if any two topologically distinguishable points in X are separated. A T1 space is also called an accessible space or a space with Fréchet topology and an R0 space is also called a symmetric space. (The term 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 also has another meaning.) A topological space is a T1 space if and only if it is both an R0 space and a Kolmogorov (or T0) space (i.e., a space in which distinct points are topologically distinguishable). A topological space is an R0 space if and only if its Kolmogorov quotient is a T1 space. If is a topological space then the following conditions are equivalent: is a T1 space. is a T0 space and an R0 space. Points are closed in ; that is, for every point the singleton set is a closed subset of Every subset of is the intersection of all the open sets containing it. Every finite set is closed. Every cofinite set of is open. For every the fixed ultrafilter at converges only to For every subset of and every point is a limit point of if and only if every open neighbourhood of contains infinitely many points of Each map from the Sierpinski space to is trivial. The map from the Sierpinski space to the single point has the lifting property with respect to the map from to the single point.