There exists a

countable set E in E such that A = [E.sup.[perpendicular to].sub.w] + A where,

Then there exists a

countable set [D.sub.0] [subset] D, such that [alpha](D) [less than or equal to] 2[alpha]([D.sub.0]).

If X is a

countable set, then a sequence [[mu].sub.n] converges strongly to a limit [mu] if and only if

Therefore, if f is replaced by [f.sup.**], the minimization problem involving [g.sub.[alpha]] gives the same solution, except possibly a

countable set of values [alpha] where the maximum is attained (either for f or [f.sup.**]) in more than one point

(48a)-(48c), a

countable set of other solutions exists for the frequency [??] = [eta].

Then by Lemma 1, there exists an open subset V containing x and a

countable set C such that V - C [subset not equal to] sInt(B).

Then, the set of all discontinuity points of f is at most a

countable set in R.

Let V = {[x.sub.1], [x.sub.2], [X.sub.3], ...} be a

countable set of variables.

Then there exists a

countable set of indices {[[alpha].sub.n]: n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using methods from linear functional analysis Conca, Planchard and Vanninathan [3] proved that this problem has a

countable set of eigenvalues which are positive and real and which converge to infinity.

What is used is the corollary that every

countable set of reals is representable in a constructible model.

We can assume without loss of generality f is one-to-one off of a

countable set [St]; also, by [A13] and [P2] we know that f extends continuously to U.

Now, let us consider a

countable set {[E.sub.i]} of subsets of diameter at most [epsilon] that covers W; that is,