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Polish spaces are separable completely metrizable spaces
Definition
A topological space is Polish when it is separable (Separability: the existence of an at most countable dense subset) and completely metrizable: its topology is induced by some complete metric (Complete metrizability: admitting a topologically equivalent complete metric is preserved by homeomorphism and by closed subspaces, and has it without being complete). No particular compatible complete metric or countable dense subset is part of the structure.
Depends on
Used by
- Under the Axiom of Choice, the Hilbert cube is compact, Polish, and universal for separable metrizable spaces Example
- For completely metrizable spaces, the separable and second-countable definitions of Polish space agree under countable choice Proposition
- Under Dependent Choice, a subspace of a Polish space is Polish exactly when it is G_δ Theorem
- Under Dependent Choice, every nonempty Polish space is a continuous image of Baire sequence space Theorem
- Under the Axiom of Countable Choice, Baire sequence space is Polish, and its standard ultrametric is complete Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 76 results over 16 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
- 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)