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# Axiom of dependent choice

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 Title: Axiom of dependent choice Author: World Heritage Encyclopedia Language: English Subject: Collection: Axiom of Choice Publisher: World Heritage Encyclopedia Publication Date:

### Axiom of dependent choice

In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice (AC) that is still sufficient to develop most of real analysis.

## Contents

• Formal statement 1
• Use 2
• Equivalent statements 3
• Relation with other axioms 4
• Footnotes 5
• References 6

## Formal statement

The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence (xn) in X such that xnRxn+1 for each n in N. (Here an entire binary relation on X is one such that for each a in X there is a b in X such that aRb.) Note that even without such an axiom we could form the first n terms of such a sequence, for any natural number n; the axiom of dependent choice merely says that we can form a whole sequence this way.

If the set X above is restricted to be the set of all real numbers, the resulting axiom is called DCR.

## Use

DC is the fragment of AC required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step.

## Equivalent statements

DC is (over the theory ZF) equivalent to the statement that every (nonempty) pruned tree has a branch. It is also equivalent to the Baire category theorem for complete metric spaces.

## Relation with other axioms

Unlike full AC, DC is insufficient to prove (given ZF) that there is a nonmeasurable set of reals, or that there is a set of reals without the property of Baire or without the perfect set property.

The axiom of dependent choice implies the Axiom of countable choice, and is strictly stronger.

## Footnotes

1. ^ Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 933--934.