Distinguishing Points

Topology · Axiom Academy

How do we tell points apart in a topological space? Imagine you're looking at two points in a space. In everyday life, we tell points apart by their coordinates, distances, or visual position. But in topology, we have a much more fundamental question: Let's visualize this. Below are two points, x and y , sitting in a topological space. Watch as we try to find open sets that contain one point but not the other. T₀ Spaces: The Weakest Separation The weakest way to distinguish points is called the T₀ axiom (or Kolmogorov separation). It requires that for any two distinct points, there exists at least one open set containing one point but not the other. A topological space is called T₀ if for any two distinct points , there exists an open set such that either and , or and . Notice the key word: "or". We only need one open set to separate in one direction . The visualization below shows this asymmetric separation. T₁ Spaces: Symmetric Separation The T₀ axiom feels incomplete—what if we want separation to work both ways ? That's exactly what the T₁ axiom provides. For any two distinct points, we can find open sets that separate in both directions . A topological space is called T₁ if for any two distinct points , there exists an open set containing but not , and there exists an open set containing but not . Now we have two-way separation! Each point can be "shielded" from the other using its own open set.

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