T1 Spaces

Topology · Axiom Academy

Understanding when points can be separated by open sets In topology, we often want to distinguish between different points in a space. The T1 separation axiom provides a basic level of separation: any two distinct points can be "separated" in the sense that each has an open neighborhood not containing the other. A topological space is called a T1 space (or Fréchet space ) if for every pair of distinct points , there exist open sets and such that: The T1 axiom has a beautiful equivalent characterization that's often easier to verify in practice. A space is T1 if and only if every singleton set is closed. Let's look at some concrete examples to build intuition: Any space with the discrete topology is T1. In fact, every subset is both open and closed, including all singleton sets. For any two distinct points , we can take and . with the standard topology is T1. For any , the singleton is closed (it's the complement of the open set ). This generalizes to for all . On an infinite set , the cofinite topology (where open sets are empty or have finite complement) is T1. Each singleton is closed because its complement is open (being cofinite). Non-Examples: Spaces that are NOT T1 Understanding what fails to be T1 is just as important: Any space with more than one point and the indiscrete topology is NOT T1. The only open sets are and , so we cannot find an open set containing one point but not another.

This is the written version of the interactive lesson above. See the full Topology course.