Extending mathematical structures from X to P^*(X), and some equivalents of the axiom of choice
Subject Areas : Statistics
1 -
Keywords: Equivalent statements, Binary operation, Metric, Partial order, Equivalence relation,
Abstract :
Let be a nonempty set, and be the set of all nonempty subsets of . In this paper we show that the existence of a choice function on , which is a consequence of the axiom of choice, makes it possible to extend some mathematical structures defined on to similar ones on . For instance, we show that every binary operation on can be extended to one on , and any metric defined on can be extended to a pseudo-metric on . In this way, we obtain some new equivalents of the axiom of choice. Such results are against our usual observations in mathematics, because in usual, structures and properties are inherited from a set to its subsets. As one of the important results of this paper, we present a statement whose conjunction with the principle implies the axiom of choice. Finally, we obtain two topological consequences of the axiom of choice.
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