Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Under choice, is a well-defined cardinal
Statement
Assuming choice, the collection of cardinalities of bases for is nonempty and has a least member. Hence is well-defined.
Facts & Assumptions
Given: A topological space and the definition of (Under choice, weight , density , local character , and character as raw cardinal minima and a supremum).
Under choice every set has a cardinality (Cardinal (initial ordinal) and cardinality, The well-ordering theorem).
Every nonempty set of ordinals, and hence every nonempty set of cardinals, has a least member; this is a theorem of ZF (Trichotomy and well-ordering of the ordinals).
Proof
By [A1], every basis has a cardinality. The topology of is itself a basis, so the set of cardinalities of bases is nonempty.
By [L1], its nonempty collection of cardinal values has a least member; that member is exactly the minimum in the definition of .
Depends on
Used by
- Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the ℵ₀ threshold Corollary
- Under choice, c(X)≤ d(X)≤ w(X) and χ(X),L(X)≤ w(X) Theorem
- Weight and character of the Kojman-Shelah scale subspace Theorem
Cited to discharge well-definedness by Under choice, weight w(X), density d(X), local character χ(x,X), and character χ(X) as raw cardinal minima and a supremum.
Dependency tree · two levels
24 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. H. Fremlin, Measure Theory, Chapter 5A (standard reference, not scraped)