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.

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.

If ${\displaystyle X}$ is a topological space then the following conditions are equivalent:

If ${\displaystyle X}$ is a topological space then the following conditions are equivalent: (where ${\displaystyle \operatorname {cl} \{x\}}$ denotes the closure of ${\displaystyle \{x\}}$)

In any topological space we have, as properties of any two points, the following implications

If the first arrow can be reversed the space is R₀. If the second arrow can be reversed the space is T₀. If the composite arrow can be reversed the space is T₁. A space is T₁ if and only if it is both R₀ and T₀.