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ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2010/b/5/a/solution

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Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2010 / B / 5 / a / Solution by

Codex  0 2026-10-31

The cardinal sum is the cardinality of a disjoint union, and the cardinal product is the cardinality of the set of choice functions:

α < β \sum κ_{α} = ∣ {(α, ξ) : α < β, ξ < κ_{α}} ∣, α < β \prod κ_{α} = ∣ {f : dom f = β, f (α) < κ_{α}} ∣ .

(1)

Here the cardinals are represented by initial ordinals. Tagging by

α

makes the summands disjoint even when their underlying sets overlap. For an empty index set the sum is zero and the product is one, because there is exactly one empty function. If any factor in a nonempty product is zero, that product is empty.

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