Alphabeta Math

Hausdorff via the Diagonal: Examples and Counterexamples

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1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

The diagonal of ℝ is closed in ℝ², computed from the product basis

Example

The cofinite topology on an infinite set, and the cocountable topology on ℝ, are T₁ with a diagonal whose closure is the whole square; on a countably infinite set the cocountable topology is discrete instead

Example

The graph of a continuous f : ℝ → ℝ is closed in ℝ²

Counterexample

Refuted: a function into a Hausdorff space whose graph is closed is continuous. The function equal to 1/x off 0 and to 0 at 0 has a closed graph, is discontinuous at 0 alone, and has a Hausdorff codomain

Example

Two continuous maps ℝ → ℝ agreeing at every rational are equal

Counterexample

Refuted: the agreement set of two continuous maps is closed, with no hypothesis on the codomain. Two continuous maps ℝ → {a,b} into the indiscrete two-point space have agreement set ℚ

Example

A finite Hausdorff space is discrete, and its diagonal is closed for the trivial reason that every subset of the square is

Example

A finite subset of any space is compact, so the compact separation clauses specialise to separating a point from a finite set in a Hausdorff space

Sources