Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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For completely metrizable spaces, the separable and second-countable definitions of Polish space agree under countable choice

Statement

Assume the Axiom of Countable Choice. For a completely metrizable space, separability is equivalent to second countability. Thus either countability convention gives the same notion of Polish space.

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

A topological space is Polish when it is separable (def-separable-space) and completely metrizable: its topology is induced by some complete metric (lem-complete-remetrisation). No particular compatible complete metric or countable dense subset is part of the structure. (Polish spaces are separable completely metrizable spaces).

[F2]

Assuming ACω, a metrizable space is second countable iff it is separable iff it is Lindelöf. (Assuming countable choice, a metrizable space is second countable if and only if it is separable if and only if it is Lindelöf).

Proof

technique · direct
1.1

A completely metrizable space is metrizable.

givenF1F2
2.1

Apply the published metrizable-space equivalence between separability and second countability under countable choice, without strengthening the choice claim to ZF.

step 1.1F1F2
3.1

The preceding construction and implications establish the assertion.

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 64 results over 12 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