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The Claim

If a topological space X is second countable, then every open cover of X has a countable subcover (i.e., X is Lindelöf).

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The Short Version

In standard topology, this is a correct theorem: every second-countable space is Lindelöf. The usual proof uses a countable base and chooses one cover element for each basis element, yielding a countable subcover. A specialized caveat is that in bare ZF set theory, without suitable choice principles, this implication requires extra care and may fail.

Caveats

  • The standard proof relies on ordinary set-theoretic assumptions; in bare ZF, the implication may require countable choice or may not hold.
  • The converse is false: a Lindelöf space need not be second countable.
  • In metric spaces, second countable and Lindelöf are equivalent, but outside metric spaces only the stated direction is generally valid.

The Receipts

  1. Section 30. The Countability Axioms

    East Tennessee State University

  2. second-countable spaces are Lindelöf

    nLab

  3. Second-countable implies Lindelof

    Topospaces (Subwiki)

  4. Metric Space is Lindelöf iff Second-Countable

    ProofWiki

  5. Second-Countable Space is Lindelöf

    ProofWiki

  6. A Comparison of Lindelöf-type Covering Properties of Topological Spaces

    Rose-Hulman Institute of Technology (RHIT Scholar)

  7. M 453 – Solutions 10

    Cornell University

  8. Is every Lindelöf space second countable?

    MathOverflow

  9. Section 30: The Countability Axioms

    dbFin

  10. Second Countable and Lindelöf

    PlanetMath

+ 13 more sources — see the full list on Lenz

Filed Under

Lindelöf SpaceOpen CoverSecond Countable Space

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