Separability: the existence of an at most countable dense subset

Depends on

• Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets
• Finite, countably infinite, countable, uncountable

Used by

• Geometric median of block means Algorithm
• A separable infinite-dimensional Hilbert space is ℓ² Corollary
• A separable predual has weak-star sequentially compact dual ball Corollary
• c₀ is not isomorphic to a dual space Corollary
• Compact groups are type I and their direct integrals collapse to discrete Hilbert sums Corollary
• Separable reflexive space has separable dual Corollary
• Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the ℵ₀ threshold Corollary
• A nonconstant summary can lose an identifiable parameter Counterexample
• A nonintegrable error envelope destroys the linear Hellinger rate Counterexample
• An identifiable model can have persistent ABC summary distortion Counterexample
• Infinite-dimensional Gaussian random-walk reverse flow is singular Counterexample
• Pointwise continuity alone does not guarantee the zero-tolerance limit Counterexample
• Positive ABC tolerance changes an identity-summary target Counterexample
• Weakly measurable need not be strongly measurable Counterexample
• A Borel prior concentrated on a forward-map domain Definition
• Centered Borel Gaussian random element on a separable Banach space Definition
• L-spaces, S-spaces, and strong S-spaces Definition
• L2-m-approximable causal Bernoulli shifts Definition
• Mackey Borel structure and countable separation of the unitary dual Definition
• P-ideals, PID, the pseudointersection number, and S-spaces Definition
• Polish spaces are separable completely metrizable spaces Definition
• Small-ball MAP conventions and Cameron–Martin relative modes Definition
• Spectral multiplicity function in the separable case Definition
• Systems of imprimitivity for a Borel G-space Definition
• Wiener measure on continuous path space Definition
• A Bernoulli count is a sufficient summary for a finite-sample posterior Example
• A finite deterministic copy has mutual information equal to entropy Example
• A Gaussian wavelet series at the Besov threshold Example
• A polynomial-tail law with finite variance Example
• A stable two-point potential gives a Lipschitz posterior Example
• A trace-class Gaussian with full support and a proper Cameron–Martin space Example
• Assuming countable choice, ω₁ is first countable and countably compact but is not separable or Lindelöf Example
• Canonical least-ball selection removes choice Example
• Changing a conditional kernel on a null fiber leaves information unchanged Example
• Coarsening the conditioning variable can increase conditional mutual information Example
• Diagonal trace-class Fredholm determinant Example
• Exponential singular-value decay and an exponentially weighted truth class Example
• Multiplicity-two diagonal representation Example
• One corrupted block is rejected by the geometric median Example
• Polynomial singular-value decay and a power Gaussian prior Example

…and 94 more results.

Dependency tree · two levels

10 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

• UCR General Topology Notes (standard reference, not scraped)
• Separable space (Wikipedia) (standard reference, not scraped)