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

• The canonical white-noise law is tight under the Axiom of Choice Corollary
• A nonconstant summary can lose an identifiable parameter Counterexample
• A nontransitive system with two orbits is not classified by one stabilizer Counterexample
• An identifiable model can have persistent ABC summary distortion Counterexample
• Pointwise continuity alone does not guarantee the zero-tolerance limit Counterexample
• Positive ABC tolerance changes an identity-summary target Counterexample
• A Borel prior concentrated on a forward-map domain Definition
• Analytic and coanalytic sets by closed projection Definition
• Borel probability law on a polish space Definition
• Centered Borel Gaussian random element on a separable Banach space Definition
• Standard Borel spaces Definition
• The Euclidean quadratic Wasserstein space Definition
• A Bernoulli count is a sufficient summary for a finite-sample posterior Example
• A dominated posterior on a nonstandard-Borel observation space Example
• A finite deterministic copy has mutual information equal to entropy Example
• A proposal with disconnected support Example
• A standard-Borel posterior kernel for countable data Example
• A three-state random-walk Metropolis kernel Example
• A trace-class Gaussian with full support and a proper Cameron–Martin space Example
• A two-state independence sampler Example
• Changing a conditional kernel on a null fiber leaves information unchanged Example
• Coarsening the conditioning variable can increase conditional mutual information Example
• Countable discrete spaces are standard borel Example
• Euclidean borel spaces are standard borel Example
• Finite transitive G-sets recover the stabilizer-induction classification Example
• Multiplicity-two diagonal representation Example
• Rejection creates a self-transition Example
• The regular translation system on L²(ℝⁿ): position, momentum and trivial stabilizer Example
• Under the Axiom of Choice, the Hilbert cube is compact, Polish, and universal for separable metrizable spaces Example
• Vanishing acceptance probability with a stable zero-tolerance target Example
• A countable discrete measurable space is standard Borel Lemma
• Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups Lemma
• Borel subspaces admit polish presentations Lemma
• C star state GNS construction, purity and Polish pure-state spaces Lemma
• Cantor and Baire sequence spaces and coordinate codings Lemma
• Closed subspaces, products, and Baire parametrization Lemma
• Closed witness codings and completion measurability of Borel projections Lemma
• Conull Borel uniformizations and Borel versions of measured suprema Lemma
• Faithful essential pure-state orbits obstruct countable separation Lemma
• Finite products of standard Borel spaces are standard Borel Lemma

…and 32 more results.

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)