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

• 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
• 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
• Weakly measurable need not be strongly measurable Counterexample
• L-spaces, S-spaces, and strong S-spaces Definition
• P-ideals, PID, the pseudointersection number, and S-spaces Definition
• Polish spaces are separable completely metrizable spaces Definition
• Spectral multiplicity function in the separable case Definition
• Wiener measure on continuous path space Definition
• Assuming countable choice, ω₁ is first countable and countably compact but is not separable or Lindelöf Example
• Canonical least-ball selection removes choice Example
• The one-point compactification of the discrete real line is compact and Lindelöf but is neither first countable nor separable Example
• Under choice, the lower-limit line is regular and separable but not second countable and therefore not metrizable Example
• FALSE: every regular space is metrizable False statement
• FALSE: Lᵖ(μ) is separable for every measure μ and every 1 ≤ p < ∞ False statement
• Refuted: every separable space is second countable False statement
• Refuted: separability is hereditary False statement
• Countable boundary null partitions of a separable metric space Lemma
• Countable compactness closes in the bidual Lemma
• Eberlein–Šmulian metrization on the relevant dual ball Lemma
• Eberlein–Šmulian separable reduction Lemma
• Every uncountable Moore subspace is nonseparable Lemma
• James nonreflexivity sequence separated from an annihilator Lemma
• Linear-order completion and density Lemma
• Maximal orthogonal family of cyclic reducing subspaces Lemma
• Under choice, if |I|>2^ℵ₀, then the Cantor cube 2^I is not separable Lemma
• Every separable space satisfies the countable chain condition Proposition
• Sequence ell-one versus nonatomic L-one for the RNP Remark
• Assuming countable choice, a metrizable space is second countable if and only if it is separable if and only if it is Lindelöf Theorem
• Assuming countable choice, every second countable space is separable Theorem
• Bing's Q-set space is a normal nonmetrizable Moore space Theorem
• Every separable metrizable space embeds in the Hilbert cube [0,1]^ℕ Theorem
• Existence of continuous Brownian motion Theorem
• If μ is sigma-finite and A is countably generated, then Lᵖ(μ) is separable for 1 ≤ p < ∞ Theorem
• Kolmogorov continuity criterion in one parameter Theorem
• L^∞[0,1] is not separable Theorem
• Multiplication operator form of the bounded normal spectral theorem Theorem
• Pettis measurability criterion for strong measurability Theorem

…and 6 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)