TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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A ZFC L-space

Statement

ZFC proves that $(ω_{1}, τ [ω_{1}])$ is a zero-dimensional regular Hausdorff, hereditarily Lindelöf, nonseparable space. In particular it is an L-space.

Facts & Assumptions

Given: ZFC and the fixed Moore minimal-walk construction.

[F1]

Moore's clopen-generated topology makes every Moore space zero-dimensional regular Hausdorff.

[F2]

Moore's topology is hereditarily Lindelof proves hereditary Lindelöfness for every $X \subseteq ω_{1}$ .

[F3]

Every uncountable Moore subspace is nonseparable proves nonseparability whenever $X$ is uncountable.

[F4]

L-spaces, S-spaces, and strong S-spaces defines an L-space as regular Hausdorff, hereditarily Lindelöf, and not hereditarily separable.

Proof

technique · direct assembly

1.1

Take $X = ω_{1}$ . By [F1] its Moore topology is zero-dimensional, regular, and Hausdorff; by [F2] it is hereditarily Lindelöf.

F1F2given

2.1

The set $ω_{1}$ is uncountable, so [F3] says the whole space is nonseparable. A hereditarily separable space must itself be separable, since the whole underlying set is one of its subspaces. Thus this space is not hereditarily separable.

F3givenstep 1.1

3.1

The four conclusions in steps 1.1 and 2.1 meet [F4] exactly, proving that $(ω_{1}, τ [ω_{1}])$ is an L-space. No new choice is made in this assembly; every choice-dependent construction or thinning belongs to the corresponding supplier under its own stated hypotheses. The empty and singleton cases of the topology are irrelevant to the witness because $ω_{1}$ is uncountable.

F1F2F3F4step 1.1step 2.1∎

Depends on

Used by

L-space and S-space existence is asymmetric Theorem

Dependency tree · two levels

12 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

Moore, A solution to the L space problem, Theorem 1.3 and Section 7, printed pp. 2 and 21–24 (standard reference, not scraped)