Partitions of Unity and Paracompactness

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1 · Prerequisites

Combinatorics

Finite Counting, Factorials and Binomial Coefficients

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Lemma

Locally finite families remain locally finite after taking closures, closure commutes with their union, and a locally finite union of closed sets is closed

Definition

Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined

Lemma

A locally finite family of continuous nonnegative functions has a continuous pointwise sum

Lemma

A locally finite nonnegative family with positive pointwise sum normalizes to a partition of unity

Lemma

Under choice, every open cover of a paracompact Hausdorff space has locally finite open refinements {Vₛ} and {Wₛ} with Vₛ⊆ Wₛ subseteqWₛ⊆ Uₛ

Theorem

Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity

Theorem

For a Hausdorff space, paracompactness is equivalent, under choice and dependent choice, to the existence of a locally finite subordinate partition of unity for every open cover

Corollary

Under choice and dependent choice, every open cover of a compact Hausdorff space admits a finite subordinate partition of unity

Theorem

Partitions of Unity and Paracompactness

3 · Items

Refinements, locally finite families, point-finite families, and star refinements

Locally finite families remain locally finite after taking closures, closure commutes with their union, and a locally finite union of closed sets is closed

Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word

Every compact space is paracompact

Every closed subspace of a paracompact space is paracompact

Every paracompact Hausdorff space is regular

Every paracompact Hausdorff space is normal

Locally finite partitions of unity and subordination to an open cover

Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined

A locally finite family of continuous nonnegative functions has a continuous pointwise sum

A locally finite nonnegative family with positive pointwise sum normalizes to a partition of unity

Under choice, every open cover of a paracompact Hausdorff space has locally finite open refinements {Vₛ} and {Wₛ} with Vₛ⊆ Wₛ subseteqWₛ⊆ Uₛ

Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity

For a Hausdorff space, paracompactness is equivalent, under choice and dependent choice, to the existence of a locally finite subordinate partition of unity for every open cover

Under choice and dependent choice, every open cover of a compact Hausdorff space admits a finite subordinate partition of unity

Stone's theorem, under choice: every metric space is paracompact

Under choice and dependent choice, metric open covers admit locally finite subordinate partitions of unity

Assuming countable choice, every countably compact paracompact Hausdorff space is compact

Under countable choice, every regular Lindelöf space is paracompact

Refuted: every paracompact space is normal

Assuming choice, refuted: paracompactness is hereditary

Assuming choice, refuted: paracompactness is productive

Choice and convention ledger for paracompactness, Stone's theorem, and partitions of unity

Partitions of Unity and Paracompactness

3 · Items

Refinements, locally finite families, point-finite families, and star refinements

Locally finite families remain locally finite after taking closures, closure commutes with their union, and a locally finite union of closed sets is closed

Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word

Every compact space is paracompact

Every closed subspace of a paracompact space is paracompact

Every paracompact Hausdorff space is regular

Every paracompact Hausdorff space is normal

Locally finite partitions of unity and subordination to an open cover

Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined

A locally finite family of continuous nonnegative functions has a continuous pointwise sum

A locally finite nonnegative family with positive pointwise sum normalizes to a partition of unity

Under choice, every open cover of a paracompact Hausdorff space has locally finite open refinements {Vₛ} and {Wₛ} with Vₛ⊆ Wₛ subseteqWₛ⊆ Uₛ

Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity

For a Hausdorff space, paracompactness is equivalent, under choice and dependent choice, to the existence of a locally finite subordinate partition of unity for every open cover

Under choice and dependent choice, every open cover of a compact Hausdorff space admits a finite subordinate partition of unity

Under choice, every open cover of a metric space has a point-finite open refinement

Under choice, Ornstein's second construction turns a point-finite metric open cover into a locally finite open refinement

Stone's theorem, under choice: every metric space is paracompact

Under choice and dependent choice, metric open covers admit locally finite subordinate partitions of unity

Assuming countable choice, every countably compact paracompact Hausdorff space is compact

Under countable choice, every regular Lindelöf space is paracompact

Refuted: every paracompact space is normal

Assuming choice, refuted: paracompactness is hereditary

Assuming choice, refuted: paracompactness is productive

Choice and convention ledger for paracompactness, Stone's theorem, and partitions of unity