T₀, T₁ and Hausdorff separation

Try

In the two-point space with opens empty, singleton one, and the whole space, one point has a smaller neighborhood than the other. An open set distinguishes them in one direction, but the two points cannot be separated symmetrically.

The T₀, T₁ and Hausdorff axioms record increasing strengths of point separation. They are independent of how a finite diagram happens to place the points on paper.

Predict

Predict which axiom says every singleton is closed. It is T₁, not merely T₀. Predict whether T₁ guarantees disjoint neighborhoods for distinct points. The cofinite topology on an infinite set disproves that implication.

Every nonempty cofinite open set misses only finitely many points. Two such opens cannot be disjoint in an infinite set, despite every singleton being closed.

Experiment

Compare the standard examples.

Space	T₀	T₁	Hausdorff
Two-point indiscrete	No	No	No
Two-point asymmetric example	Yes	No	No
Infinite cofinite space	Yes	Yes	No
Metric space	Yes	Yes	Yes

The finite T₁ case is exceptional: every finite T₁ space is discrete. This is why a three-point widget cannot exhibit a T₁ non-Hausdorff example.

Understand

A space is T₀ if two distinct points can be distinguished by an open set containing one but not the other. It is T₁ if for each ordered pair of distinct points there is an open neighborhood of the first missing the second. It is Hausdorff, or T₂, if distinct points have disjoint open neighborhoods.

T₁ is equivalent to closed singletons. For a fixed x, the T₁ neighborhoods of all other points missing x have union X without x, making that complement open. Conversely, the complement of a closed singleton containing y is an open neighborhood of any different x missing y. Hausdorff implies T₁ because disjoint neighborhoods exclude the opposite point; T₁ implies T₀ immediately.

In a finite T₁ space, every subset is a finite union of closed singletons, hence closed. Its complement is therefore open, so the topology is discrete. The cofinite infinite example cannot be replicated on finitely many points.

Hausdorff separation guarantees unique sequence limits. If a sequence converged to distinct x and y, choose disjoint neighborhoods. Eventually the sequence must be in each; after both thresholds it would lie in their empty intersection. The same proof works for nets by directedness. The converse based only on sequences can fail in general spaces, so uniqueness of sequence limits should not be used as the definition of Hausdorffness.

Apply

Verify the infinite cofinite topology is a topology: arbitrary unions of nonempty cofinite sets are cofinite because their complements lie inside one finite complement, and finite intersections have finite complementary unions. Then prove it is T₁ but not Hausdorff.

Repair “Different points always have different neighborhoods.” That is a T₀ property, not an axiom of a topological space. The indiscrete two-point example gives different points with exactly the same open neighborhoods.

Continue

Regularity and normality separate points from closed sets and pairs of closed sets, adding another layer of structure. Optional depth: prove a subspace of a Hausdorff space is Hausdorff by intersecting two disjoint ambient neighborhoods with the subspace. State explicitly why the resulting neighborhoods still contain the chosen points.