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Under the Axiom of Countable Choice, countable products and subspaces of Polish spaces are Polish
Statement
Assume the Axiom of Countable Choice. A countable product of Polish spaces is Polish, and every subspace of a Polish space is Polish.
Facts & Assumptions
Given: The objects, hypotheses, and choice principles stated above.
Assume the Axiom of Countable Choice. Every countable product of completely metrizable spaces is completely metrizable, including the empty product. (Under countable choice, a countable product of completely metrizable spaces is completely metrizable).
Assuming , a countable product of second countable spaces is second countable. (Assuming countable choice, a countable product of second countable spaces is second countable).
Assume the Axiom of Countable Choice. A subspace of a Polish space is Polish if and only if it is a subset. (Under Dependent Choice, a subspace of a Polish space is Polish exactly when it is ).
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. (For completely metrizable spaces, the separable and second-countable definitions of Polish space agree under countable choice).
Proof
Use the product theorem for complete metrizability and the published theorem that countable products of second-countable spaces are second countable.
For a subspace apply the Polish-subspace characterisation.
The preceding construction and implications establish the assertion.
Depends on
- Under countable choice, a countable product of completely metrizable spaces is completely metrizable
- Assuming countable choice, a countable product of second countable spaces is second countable
- Under Dependent Choice, a subspace of a Polish space is Polish exactly when it is $G_\delta$
- For completely metrizable spaces, the separable and second-countable definitions of Polish space agree under countable choice
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)