False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: the underlying-set functor from topological spaces is monadic

Statement

False claim: the underlying-set functor $U : T o p \to S e t$ is monadic.

Facts & Assumptions

Given: The underlying-set functor on topological spaces.

[L1]

Every monadic functor reflects isomorphisms (Every monadic functor is conservative).

[L2]

A continuous bijection need not be a homeomorphism; the published false statement supplies a two-point witness (FALSE: every continuous bijection of topological spaces is a homeomorphism).

[L3]

Topological spaces and continuous maps form the category $T o p$ (Topological spaces and continuous maps form the large locally small category $T o p$ ).

Refutation

technique · direct

1.1

By [L1], conservativity is necessary for the underlying-set functor to be monadic.

1.2

On $S = {a, b}$ , let $S_{d}$ be discrete and let $S_{s}$ have the Sierpiński topology ${\emptyset, {b}, S}$ . The identity function $q : S_{d} \to S_{s}$ is a continuous bijection, but its inverse is not continuous because ${a}$ is open in $S_{d}$ and not in $S_{s}$ . This is the failure recorded in [L2] inside the category [L3].

L2L3

2.1

The underlying function $U q$ is an isomorphism in $S e t$ , while $q$ is not an isomorphism in $T o p$ because it is not a homeomorphism. Thus $U$ does not reflect this isomorphism.

step 1.2algebra

3.1

The functor $U$ is not conservative by step 2.1 and therefore cannot be monadic by step 1.1.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. Mehrle, Category Theory Part III, Example 5.20(e) (standard reference, not scraped)
E. Riehl, Category Theory in Context, 2nd ed., Section 5.6 (standard reference, not scraped)