LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Under countable choice, every regular Lindelöf space is paracompact

Statement

Assume the Axiom of Countable Choice. Every regular Lindelöf topological space is paracompact.

Facts & Assumptions

Given: Countable choice, a regular Lindelöf space $X$ , and an open cover $U$ .

[A1]

Countable choice selects from a countably indexed family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L1]

If $x \in U$ with $U$ open in a regular space, then some open $V$ satisfies $x \in V \subseteq \overline{V} \subseteq U$ (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]

Lindelöfness gives an at most countable subcover, and paracompactness asks for a locally finite open refinement (Countably compact, Lindel"of, sequentially compact, limit point compact and $σ$ -compact spaces, and relatively compact subsets, Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word).

Proof

technique · constructive

1.1L1F1construct

The family of all open $V$ for which $\overline{V} \subseteq U$ for some $U \in U$ covers $X$ by [L1]; by Lindelöfness take a sequence $V_{0}, V_{1}, \dots$ covering $X$ .

2.1A1step 1.1choose

By [A1], choose $U_{n} \in U$ with $\overline{V_{n}} \subseteq U_{n}$ for each $n$ .

3.1step 2.1construct

Put $W_{n} : = U_{n} ∖ ⋃_{i < n} \overline{V_{i}}$ . Each $W_{n}$ is open and lies in $U_{n}$ .

4.1step 1.1step 2.1step 3.1

The $W_{n}$ cover $X$ : if $n$ is the least index with $x \in U_{n}$ , then $x \notin \overline{V_{i}}$ for $i < n$ , since $\overline{V_{i}} \subseteq U_{i}$ , and hence $x \in W_{n}$ .

4.2step 3.1

The cover is locally finite: for $x \in V_{k}$ , the neighbourhood $V_{k}$ is disjoint from $W_{n}$ for every $n > k$ , while it can meet only $W_{0}, \dots, W_{k}$ .

5.1F1step 3.1step 4.1step 4.2discharge-construct∎

Thus ${W_{n}}$ is a locally finite open refinement of $U$ , and [F1] proves paracompactness.

Under countable choice, every regular Lindelöf space is paracompact

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources