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Assuming countable choice, every second countable space is Lindelöf
Statement
Assuming , every second countable space is Lindelöf.
Facts & Assumptions
Given: A countable basis and an open cover of .
Countable choice selects from the nonempty families indexed by the eligible basis members (The Axiom of Countable Choice ()).
Proof
For each that lies in some , use [A1] to select one such .
The selected family is countable and covers : a point lies in a cover member, and a basis member containing it lies inside that member.
Thus every open cover has a countable subcover, so is Lindelöf.
Depends on
Used by
- The holomorphic line bundle associated to a divisor Definition
- A C² first-integral period annulus has a C² leaf product Lemma
- A C² leaf meets a local box transversal in at most countably many points Lemma
- A countable coordinate-bump map embeds a manifold in countable Euclidean data Lemma
- A dipole Green function exists on a Riemann surface Lemma
- A universal covering of a Riemann surface inherits a unique complex structure Lemma
- Complex flag splitting with injective real pullback on smooth bases Lemma
- Every open cover of a manifold has a countable relatively compact coordinate-ball subcover Lemma
- L1 of a second-countable locally compact group is separable Lemma
- Local analytic separation and saturated Borel quotient images Lemma
- Oriented real two-plane splitting with real-cohomology injection Lemma
- Smooth boundary tests separate locally integrable traces Lemma
- Smooth manifolds have CW homotopy type Lemma
- Implication, preservation, counterexample, and choice ledger for the countability axioms 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, the cotangent bundle has a canonical smooth 2n-manifold structure Theorem
- Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure Theorem
- Existence and uniqueness of maximal connected integral manifolds Theorem
- Smooth partitions of unity exist on manifolds with boundary Theorem
- Topological manifolds are metrizable and paracompact Theorem
Dependency tree · two levels
20 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
- nLab: second-countable spaces are Lindelöf (standard reference, not scraped)
- UCR General Topology Notes (standard reference, not scraped)