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
• 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
• Countable base Banach manifold and smooth map Definition
• Topological manifolds with boundary Definition
• Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces 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
• CH makes the nice refinement strongly hereditarily separable Lemma
• The germ neighborhoods form a Hausdorff, second-countable Riemann-surface atlas Lemma
• Countable disjoint unions of fixed-dimensional smooth manifolds are smooth manifolds Proposition
• Open subsets of Euclidean space have the standard smooth structure Proposition
• Products of smooth manifolds have a canonical product smooth structure Proposition
• Second countability is hereditary Proposition
• The equality B(X) tensor B(Y) = B(X x Y) needs second countability Remark
• Assuming countable choice, a countable product of second countable spaces is second countable Theorem
• 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 Lindelöf Theorem
• Assuming countable choice, every second countable space is separable Theorem
• Every second countable space is first countable Theorem
• The Borel product of Rᵐ and Rⁿ is the Borel sigma-algebra of Rᵐ⁺ⁿ Theorem

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)