TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Every paracompact Hausdorff space is normal

Statement

Every paracompact Hausdorff topological space is normal. No choice principle is used.

Facts & Assumptions

Given: A paracompact Hausdorff space $X$ and disjoint closed subsets $E, F \subseteq X$ .

[L1]

The space $X$ is regular (Every paracompact Hausdorff space is regular).

[L2]

A locally finite family commutes with closure under union (Locally finite families remain locally finite after taking closures, closure commutes with their union, and a locally finite union of closed sets is closed).

[F1]

Paracompactness supplies a locally finite open refining cover (Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word).

[F2]

Normality is separation of disjoint closed sets 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

1.1L1construct

For each $x \in E$ , regularity supplies an open $U$ containing $x$ with $\overline{U} \cap F = \emptyset$ ; therefore the family of all such $U$ , together with $X ∖ E$ , is an open cover $U$ of $X$ .

2.1F1step 1.1chooseconstruct

Take a locally finite open cover $W$ refining $U$ , and set $H : = ⋃ {W \in W : W \cap E \neq \emptyset}$ .

3.1step 1.1step 2.1

The open set $H$ contains $E$ , because a member of $W$ containing a point of $E$ cannot lie inside $X ∖ E$ .

3.2step 1.1step 2.1L2

Every member $W$ used in $H$ lies in one of the eligible $U$ and so has $\overline{W} \cap F = \emptyset$ ; hence $\overline{H} = ⋃ \overline{W}$ is disjoint from $F$ by [L2].

4.1step 3.1step 3.2F2∎

The open sets $H$ and $X ∖ \overline{H}$ contain $E$ and $F$ respectively and are disjoint, so [F2] proves normality.

Every paracompact Hausdorff space is normal

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources