A Weak Countable Choice Principle

A weak choice principle is introduced that is implied both by countable choice and by the law of excluded middle. This principle su ces to prove that metric independence is the same as linear independence in an arbitrary normed space over a locally compact eld, and to prove the fundamental theorem of algebra. 1 Bishop's principle Without appeal to the law of excluded middle, Bishop [1, Lemma 7, page 177] showed that if Y is a nonempty, complete, located subset of a metric space, and x 6= y for each y in Y , then x is bounded away from Y . In fact, he constructed, for any point x, a point y0 in Y such that if x 6= y0, then d(x; Y ) > 0. In the proof, Bishop tacitly uses countable choice, possibly even dependent choice. Bishop's construction suggests the following de nition: Y is strongly re ective if for each x there exists y0 in Y such that if x 6= y0, then x is bounded away from Y . Then Bishop's construction shows Bishop's principle: a nonempty, complete, located subset of a metric space is strongly re ective. From Bishop's principle it follows that if k is a locally compact eld, then any two norms on k are equivalent (see [3, Theorem XII.4.2]). Equivalently, metric independence and linear independence are the same in any normed space over k. Using the law of excluded middle, it is easy to show that any nonempty closed subset of a metric space is strongly re ective: let y0 = x if x is in Y , and let y0 be any