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Under choice and dependent choice, metric open covers admit locally finite subordinate partitions of unity
Statement
Assume the Axiom of Choice and the Axiom of Dependent Choice. Every open cover of a metric space admits a locally finite partition of unity subordinate to it.
Facts & Assumptions
Given: Choice, dependent choice, a metric space , and an open cover of its metric topology.
The space is paracompact under choice (Stone's theorem, under choice: every metric space is paracompact).
Every metrizable space is Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not).
A paracompact Hausdorff space has a subordinate partition of unity under choice and dependent choice (Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity).
Proof
By [L1] and [L2], is paracompact and Hausdorff.
Applying [L3] to the given cover yields the required locally finite subordinate partition of unity.
Depends on
- Stone's theorem, under choice: every metric space is paracompact
- Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity
- Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
Used by
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Sources
- J. Robbin, Partitions of Unity (standard reference, not scraped)
- D. Ornstein, A New Proof of the Paracompactness of Metric Spaces, Proc. Amer. Math. Soc. 21 (1969), 341–342 (standard reference, not scraped)