Hausdorff via the Diagonal

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2 · Logical flowchart

3 · Items

Definition

The diagonal Δ_X ⊆ X × X, the diagonal map δ_X, and the pairing ⟨ f, g ⟩ of two maps

Lemma

δ_X is a topological embedding of X onto Δ_X, and ⟨ f, g ⟩ is continuous whenever f and g are

Theorem

A space is Hausdorff if and only if its diagonal is closed in the square carrying the product topology

Corollary

For continuous f, g : Z → Y with Y Hausdorff the agreement set { z ∈ Z : f(z) = g(z) } is closed in Z

Corollary

Two continuous maps into a Hausdorff space that agree on a dense subset are equal

Lemma

A map into a compact space whose graph is closed is continuous; so for a compact Hausdorff codomain, continuity and closedness of the graph are equivalent

Lemma

In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular

Remark

Why the criterion is about the product topology, and the choice cost of the compact separation lemmas

False statement

FALSE: every function between topological spaces whose graph is closed in the product is continuous

False statement

Hausdorff via the Diagonal

3 · Items

The diagonal Δ_X ⊆ X × X, the diagonal map δ_X, and the pairing ⟨ f, g ⟩ of two maps

δ_X is a topological embedding of X onto Δ_X, and ⟨ f, g ⟩ is continuous whenever f and g are

A space is Hausdorff if and only if its diagonal is closed in the square carrying the product topology

For continuous f, g : Z → Y with Y Hausdorff the agreement set { z ∈ Z : f(z) = g(z) } is closed in Z

Two continuous maps into a Hausdorff space that agree on a dense subset are equal

In a Hausdorff space a sequence converges to at most one point

A compact Hausdorff space is regular and normal, hence T₃ and T₄

The graph of a continuous map into a Hausdorff space is closed in the product

A map into a compact space whose graph is closed is continuous; so for a compact Hausdorff codomain, continuity and closedness of the graph are equivalent

In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular

Why the criterion is about the product topology, and the choice cost of the compact separation lemmas

FALSE: every function between topological spaces whose graph is closed in the product is continuous

FALSE: two continuous maps that agree on a dense subset of their common domain are equal, with no hypothesis on the codomain