Second countability: an at most countable basis for the topology

Depends on

• Basis and subbasis for a topology, and the topology generated by a family of sets
• Finite, countably infinite, countable, uncountable

Used by

• A discrete embedded submanifold is locally closed and countable Corollary
• Every Riemann surface is a quotient of a simply connected model Corollary
• Locally finite Borel measures on second-countable LCH spaces are regular Corollary
• Topological recurrence on second-countable spaces Corollary
• Under choice, every regular T₁ second-countable space is metrizable Corollary
• Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the ℵ₀ threshold Corollary
• A closed subspace of ell-infinity that is not complemented Counterexample
• An uncountable disjoint union of points is not second-countable Counterexample
• Infinite-dimensional Gaussian random-walk reverse flow is singular Counterexample
• The unitary dual need not be Hausdorff Counterexample
• Complex manifolds, holomorphic vector bundles and Hermitian metrics Definition
• Countable base Banach manifold and smooth map Definition
• Leafwise paths and leafwise homotopy relative to endpoints Definition
• Mackey Borel structure and countable separation of the unitary dual Definition
• Random measures and locally finite point processes Definition
• Riemann surfaces and holomorphic atlases Definition
• Systems of imprimitivity for a Borel G-space Definition
• Topological manifolds with boundary Definition
• Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces Definition
• Transitive systems of imprimitivity and their normalized measure class Definition
• A locally integrable density functional is represented by g dlambda Example
• A projection with finite-dimensional kernel is Fredholm Example
• A regular level set in a Banach space Example
• An uncountable discrete space is metrizable and has a discrete basis, but is not second countable Example
• Assuming choice, the lower-limit plane is first countable, separable, and ccc, but not second countable or Lindelöf Example
• Under choice, the lower-limit line is regular and separable but not second countable and therefore not metrizable Example
• An arbitrary disjoint union of second-countable manifolds need not be second-countable False statement
• FALSE: every regular space is metrizable False statement
• Refuted: every first countable space is second countable False statement
• Refuted: every separable space is second countable False statement
• A compact metric space is second-countable under countable choice Lemma
• A measurable direct integral of unitary representations is strongly continuous Lemma
• A separably acting abelian von Neumann algebra has a self-adjoint generator Lemma
• A sequential approximate identity concentrated near the identity Lemma
• A transitive Borel G-space with a quasi-invariant measure class is ergodic Lemma
• Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups Lemma
• Borel sigma-algebra of a product of second-countable spaces Lemma
• CH makes the nice refinement strongly hereditarily separable Lemma
• Degree two of ℘ and its four branch points Lemma
• Heisenberg group polarization and the fixed-central-character orbit Lemma

…and 34 more results.

Dependency tree · two levels

8 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)
• Second-countable space (Wikipedia) (standard reference, not scraped)