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Collectionwise normal Moore spaces are metrizable
Statement
In , every collectionwise normal Moore space is metrizable (Normalized families and collectionwise normality, Moore spaces and developments, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not).
Facts & Assumptions
Given: A collectionwise normal Moore space .
Collectionwise normality implies normality: for disjoint closed the two-member family is discrete, and separating it gives disjoint open sets containing and (Normalized families and collectionwise normality, Normal spaces and spaces, with the source disagreement over whether normality includes stated explicitly, Discrete families and -locally-finite and -discrete bases).
Every collectionwise normal Moore space is screenable (Collectionwise normal Moore spaces are screenable).
Every normal screenable Moore space is metrizable (Normal screenable Moore spaces are metrizable).
Proof
The space is normal by [F1] and screenable by [F2]; it is a Moore space by hypothesis.
By [F3] applied to the normal screenable Moore space , the space is metrizable.
Remarks
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This is Bing's Theorem 10 through Theorem 8. The separate screenability lemma is the combinatorial half of Bing's Theorem 10, and the metrization theorem is his Theorem 8 with the metrization criterion of Theorem 3; the two items are kept apart because the first carries the well-ordering argument and the second carries the metric construction.
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No recorded result is used. Both suppliers are items of this page, proved before this one; the argument is short only because the work sits in those two items.
Depends on
- Moore spaces and developments
- Normalized families and collectionwise normality
- Collectionwise normal Moore spaces are screenable
- Normal screenable Moore spaces are metrizable
- The Axiom of Choice
- Normal spaces and $T_4$ spaces, with the source disagreement over whether normality includes $T_1$ stated explicitly
- Discrete families and $\sigma$-locally-finite and $\sigma$-discrete bases
- Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not
Used by
Dependency tree · two levels
42 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
- R. H. Bing, Metrization of topological spaces (standard reference, not scraped)