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.
[1] An R0 space is one in which this holds for every pair of topologically distinguishable points.
(The term Fréchet space also has an entirely different meaning in functional analysis.
A finite T1 space is necessarily discrete (since every set is closed).
A space that is locally T1, in the sense that each point has a T1 neighbourhood (when given the subspace topology), is also T1.
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.