TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Under dependent choice, the unit interval is a coseparating object in compact Hausdorff spaces

Statement

Assume the Axiom of Dependent Choice. In the category of compact Hausdorff spaces, the unit interval $[0, 1]$ is a coseparating object: if $f, g : X ⇉ Y$ are distinct continuous maps, there is a continuous $h : Y \to [0, 1]$ with $h \circ f \neq h \circ g$ .

Facts & Assumptions

Given: Compact Hausdorff spaces $X, Y$ and distinct continuous maps $f, g : X ⇉ Y$ , under dependent choice.

[L1]

Every compact Hausdorff space is normal and $T_{1}$ , so singleton subsets are closed (A compact Hausdorff space is regular and normal, hence $T_{3}$ and $T_{4}$ ).

[L2]

Under dependent choice, disjoint closed subsets of a normal space are separated by a continuous map to $[0, 1]$ taking the values $0$ and $1$ on them (Urysohn's lemma, under the axiom of dependent choice: in a normal space two disjoint closed sets are separated by a continuous function into $[0, 1]$ , and conversely such a space is normal).

[L3]

A coseparating object distinguishes distinct parallel maps by postcomposition (Separating and coseparating sets of objects).

Proof

technique · direct

1.1

Since $f \neq g$ , fix $x \in X$ with $f (x) \neq g (x)$ . By [L1], the singleton sets ${f (x)}$ and ${g (x)}$ are disjoint closed subsets of the normal space $Y$ .

givenL1choose

2.1

By [L2], there is a continuous $h : Y \to [0, 1]$ with $h (f (x)) = 0$ and $h (g (x)) = 1$ . Thus $(h f) (x) \neq (h g) (x)$ , so $h f \neq h g$ and [L3] proves that $[0, 1]$ is coseparating. Both endpoints are used, and the only nonempty selection is the displayed point $x$ .

step 1.1L2L3∎

Depends on

Used by

Under the ultrafilter lemma and dependent choice, compact Hausdorff spaces satisfy the explicit SAFT hypotheses for their inclusion into topological spaces Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 78 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

S. Mac Lane, Categories for the Working Mathematician, compact-Hausdorff example after V.8 (standard reference, not scraped)
E. Riehl, Category Theory in Context, example 4.7.12 (standard reference, not scraped)