LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every regular Lindelöf space is normal

Statement

Every regular Lindelöf space is normal.

Facts & Assumptions

Given: A regular Lindelöf space $X$ and disjoint closed sets $A,B\subseteq X$ .

[L1]

If $x\in U$ and $U$ is open in a regular space, there is open $V$ with $x\in V\subseteq\overline V\subseteq U$ (A space is regular if and only if every point has a neighbourhood base of closed neighbourhoods, if and only if $x \in U$ open gives an open $V$ with $x \in V \subseteq \overline{V} \subseteq U$ ).

[F1]

Lindelöf means that every open cover has an at most countable subcover (Countably compact, Lindel"of, sequentially compact, limit point compact and $\sigma$ -compact spaces, and relatively compact subsets).

[F2]

Normality is separation of disjoint closed subsets by disjoint open sets (Normal spaces and $T_4$ spaces, with the source disagreement over whether normality includes $T_1$ stated explicitly).

Proof

technique · direct

1.1

Let $\mathcal V$ be the family of all open $V\subseteq X$ satisfying $\overline V\subseteq X\setminus B$ . For every $a\in A$ , [L1] supplies a member of $\mathcal V$ containing $a$ , so $\mathcal V\cup\{X\setminus A\}$ is an open cover of $X$ .

2.1

By Lindelöfness, an at most countable subfamily $\mathcal W\subseteq\mathcal V$ covers $A$ .

F1step 1.1

3.1

Put $U=\bigcup\mathcal W$ and $W=X\setminus\bigcup_{V\in\mathcal W}\overline V$ . Then $U,W$ are open, $A\subseteq U$ , $B\subseteq W$ , and $U\cap W=\varnothing$ .

step 1.1step 2.1

4.1

Thus $X$ is normal.

F2step 3.1∎

Depends on

Used by

Assuming countable choice, the lower-limit line is normal Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. R. Munkres, Topology, 2nd ed., §31 (standard reference, not scraped)
MSSC topology text, §16 (standard reference, not scraped)