In mathematics, the axiom of dependent choices, denoted DC, is a weak form of the axiom of choice (AC) which is still sufficient to develop most of real analysis. 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... In mathematics, a non-measurable set is a set whose structure is so complicated it sheds light on the very notion of length, area or volume. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A subset of a topological space has the property of Baire (Baire property) if it differs from an open set by a meager set; that is, if there is an open such that is meager. ... In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset. ...

The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence (x_n) in X such that x_nRx_n+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 choices merely says that we can form a whole sequence this way, which is intuitively obvious. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

If the set X above is restricted to be the set of all real numbers, the resulting axiom is called DC_R. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

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. Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ... In mathematics, a countable set is a set with the same cardinality (i. ...

DC is (over the theory ZF) equivalent to the statement that every (nonempty) pruned tree has a branch. It is also equivalent^[1] to the Baire category theorem for complete metric spaces. In descriptive set theory, a tree on a set is a set of finite sequences of elements of that is closed under subsequences. ... The Baire category theorem is an important tool in general topology and functional analysis. ...

See also

• Axiom of countable choice

The axiom of countable choice or axiom of denumerable choice is an axiom of set theory similar to the axiom of choice. ...

References

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.

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.