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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.
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).
Assuming , 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
A completely metrizable space is metrizable.
Apply the published metrizable-space equivalence between separability and second countability under countable choice, without strengthening the choice claim to ZF.
The preceding construction and implications establish the assertion.
Depends on
Used by
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Sources
- David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
- Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
- MFF General Topology course summary, §4.3 (standard reference, not scraped)
- Jesse Peterson, Real Analysis, §§3.6–3.7 (standard reference, not scraped)