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Dependent Choice is equivalent to the complete-metric Baire principle over ZF
Statement
Over ZF, the serial-relation Dependent Choice principle (equivalently, its prescribed-start form) holds if and only if every complete metric space is Baire. The principles use nonempty serial carriers and -indexed open dense families, respectively; the Baire assertion includes the empty space. This is an internal equivalence over ZF, not a consistency or independence assertion.
Facts & Assumptions
Given: ZF and the two principles in the statement.
DC implies CM-Baire with these conventions (Serial Dependent Choice implies the complete-metric Baire principle over ZF).
CM-Baire implies both starting-point-free and prescribed-start DC (The complete-metric Baire principle implies Dependent Choice over ZF).
Proof
Assume DC. The forward implication applies over ZF to every complete metric space and every -indexed sequence of open dense subsets, and says that their intersection is dense. This is exactly CM-Baire, including its empty-space instance.
Assume CM-Baire. The reverse implication applies to every nonempty set and every serial relation on it, and supplies both the unrestricted and prescribed-start chain principles. Thus CM-Baire implies DC. These two implications prove the stated equivalence over ZF.
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Sources
- Miller, Lecture notes on set theory without choice; Proposition 5.4(1) equivalent to (2), pp.10–11 (standard reference, not scraped)
- Karagila, Zornian Functional Analysis, Definition 4 and Chapter 2, pp. 4–5, 8–11 (standard reference, not scraped)