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Example text

Show also that in general (a) and (b) are not equivalent for a subset X of a nonseparable complete metric space E . NSTEIN CONSTRUCTION 29 7. Let us consider the first uncountable ordinal number wl equipped with its order topology, and let Z= (X c w , : ( 3 F c w l ) ( F is closed, curd(F) = wl, F nX = 8)). Prove that Z is a a-ideal of subsets of wl. The elements of Z are usually called nonstationary subsets of wl. Respectively, a set Z C wl is called a stationary subset of w l if Z is not nonstationary.

Proof. Let A be a subset of R described in Lemma 2. In virtue of the Kuratowski-Zorn lemma, there exists a maximal (with respect to the NONMEASURABLE SETS ASSOCIATED WITH HAMEL BASES 39 inclusion relation) rationally independent subset of A. We fix such a subset and denote it by H . Our goal is to show that H is a Hamel basis in R . Suppose to the contrary that there is an element r E R for which where l i n Q ( H )stands for the linear hull (over Q) of the set H . In view of the equality A+A=R, there are two elements a1 E A and a2 E A such that r = a1 +an.

Of course, Bernstein's argument is heavily based on the Axiom of Choice. Namely, Bernstein utilizes the fact that there exists a well ordering of the family of all uncountable closed subsets of R. The above-mentioned result of Bernstein is interesting in various respects. First of all, it admits generalizations to many other cases, where, for example, a topological space or a measure space are given and a Bernstein type subset of that space is required to be constructed (compare Exercise 5 of this chapter).