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Topological space in which all singleton sets are closed
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 separated 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 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 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.

The terms "T1", "R0", and their synonyms can also be applied to such variations of topological spaces as uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constant nets) are unique (for T1 spaces) or unique up to topological indistinguishability (for R0 spaces). As it turns out, uniform spaces, and more generally Cauchy spaces, are always R0, so the T1 condition in these cases reduces to the T0 condition. But R0 alone can be an interesting condition on other sorts of convergence spaces, such as pretopological spaces.