Polish spaces are separable completely metrizable spaces

Depends on

• Separability: the existence of an at most countable dense subset
• Complete metrizability: admitting a topologically equivalent complete metric is preserved by homeomorphism and by closed subspaces, and $(0,\infty)$ has it without being complete

Used by

• Analytic and coanalytic sets by closed projection Definition
• Borel probability law on a polish space Definition
• Standard Borel spaces Definition
• Countable discrete spaces are standard borel Example
• Euclidean borel spaces are standard borel Example
• Multiplicity-two diagonal representation Example
• Under the Axiom of Choice, the Hilbert cube is compact, Polish, and universal for separable metrizable spaces Example
• Borel subspaces admit polish presentations Lemma
• Cantor and Baire sequence spaces and coordinate codings Lemma
• Closed subspaces, products, and Baire parametrization Lemma
• Uncountable splitting in a Polish space Lemma
• Under countable choice, continuous path space is Polish Lemma
• For completely metrizable spaces, the separable and second-countable definitions of Polish space agree under countable choice Proposition
• Assuming countable choice, Borel probability measures on Polish spaces are inner regular Theorem
• Cramer wold device Theorem
• Kolmogorov construction of the canonical Gaussian process Theorem
• Levy continuity theorem converse Theorem
• Spectral multiplicity model for separably acting abelian von Neumann algebras Theorem
• 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 · two levels

28 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

• 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)