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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.
==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 (''x''''n'') in ''X'' such that ''x''''n''''R'x''''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 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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