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Stone's theorem, under choice: every metric space is paracompact
Statement
Assume the Axiom of Choice. Every metric space is paracompact.
Facts & Assumptions
Given: The Axiom of Choice, a metric space , and an arbitrary open cover of its metric topology.
Under choice, every metric open cover has a point-finite open refinement, and Ornstein's second construction turns that point-finite cover into a locally finite open refinement (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).
Paracompactness means that every open cover has such a refinement (Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word).
Proof
Apply [L1] to the arbitrary cover .
The resulting locally finite open refinement is exactly the condition in [F1], so is paracompact.
Remarks
The theorem is proved here with the Axiom of Choice as a sufficient hypothesis. No assertion is made that this is its exact set-theoretic strength.
Depends on
- 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
- Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word
- Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric
- The Axiom of Choice
Used by
- Under choice and dependent choice, metric open covers admit locally finite subordinate partitions of unity Corollary
- Under choice, the Niemytzki plane is Tychonoff and locally metrizable but not normal, paracompact, or metrizable Example
- Under choice, every metric space has a σ-locally-finite basis Lemma
- Choice and convention ledger for paracompactness, Stone's theorem, and partitions of unity Remark
- Choice ledger for Baire, Urysohn, Stone, and Tychonoff Remark
- The exact choice strength of Stone's theorem remains open Remark
- Under choice, a space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable Theorem
- Under the ultrafilter lemma and the Axiom of Choice, every completely metrizable space is Čech-complete Theorem
Dependency tree · two levels
17 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
- D. Ornstein, A New Proof of the Paracompactness of Metric Spaces, Proc. Amer. Math. Soc. 21 (1969), 341–342 (standard reference, not scraped)
- C. Good, I. J. Tree and W. S. Watson, On Stone's theorem and the axiom of choice (standard reference, not scraped)
- Topology 262 notes (California State University, Northridge) (standard reference, not scraped)