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Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the threshold
Statement
Assuming choice, is first countable iff , second countable iff , separable iff , Lindelöf iff , and ccc iff .
Facts & Assumptions
Given: A topological space and the Axiom of Choice, with the five raw cardinal functions and the named countability properties.
The minima defining , , and , and the suprema defining and , exist as cardinals (Under choice, is a well-defined cardinal, Under choice, is a well-defined cardinal, Under choice, and are well-defined cardinals, Under choice, is a well-defined cardinal, Under choice, is a well-defined cardinal).
First countability means a countable local base at every point, second countability means a countable basis, separability means a countable dense subset, ccc means that every pairwise-disjoint family of nonempty open sets is countable, and Lindelöfness means that every open cover has a countable subcover (First countable space: a countable neighbourhood base at every point, Second countability: an at most countable basis for the topology, Separability: the existence of an at most countable dense subset, The countable chain condition: every pairwise-disjoint family of nonempty open sets is at most countable, Countably compact, Lindel"of, sequentially compact, limit point compact and -compact spaces, and relatively compact subsets).
Proof
By [L1], and are the least cardinalities of a basis and a dense subset, respectively; hence and say exactly that such a basis and such a dense subset are at most countable.
By [L1], says that every open cover has a subcover of at most countable cardinality, and says that every pairwise-disjoint family of nonempty open sets is at most countable.
Since , one has exactly when every point has a local base of cardinality at most .
The descriptions in steps 1.1, 1.2 and 1.3 are precisely the definitions in [L2], so they yield the five asserted equivalences.
Depends on
- First countable space: a countable neighbourhood base at every point
- Second countability: an at most countable basis for the topology
- Separability: the existence of an at most countable dense subset
- The countable chain condition: every pairwise-disjoint family of nonempty open sets is at most countable
- Countably compact, Lindel\"of, sequentially compact, limit point compact and $\sigma$-compact spaces, and relatively compact subsets
- Under choice, $w(X)$ is a well-defined cardinal
- Under choice, $d(X)$ is a well-defined cardinal
- Under choice, $\chi(x,X)$ and $\chi(X)$ are well-defined cardinals
- Under choice, $L(X)$ is a well-defined cardinal
- Under choice, $c(X)$ is a well-defined cardinal
Used by
Dependency tree · two levels
36 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)