Uniformization (Set Theory)
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if R is a subset of Xtimes Y, where X and Y are Polish spaces, then there is a subset f of R that is a partial function from X to Y, and whose domain (in the sense of the set of all x such that f(x) exists) equals {xin X exists yin Y (x,y)in R},Such a function is called a uniformizing function for R, or a uniformization of R.To see the relationship with the axiom of choice, obs...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if R is a subset of Xtimes Y, where X and Y are Polish spaces, then there is a subset f of R that is a partial function from X to Y, and whose domain (in the sense of the set of all x such that f(x) exists) equals {xin X exists yin Y (x,y)in R},Such a function is called a uniformizing function for R, or a uniformization of R.To see the relationship with the axiom of choice, observe that R can be thought of as associating, to each element of X, a subset of Y. A uniformization of R then picks exactly one element from each such subset, whenever the subset is nonempty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to AC.
