Any

countable set in [epsilon] containing E also forms a refiner set of A, then we may also assume that y[[zeta].sub.n]'s are all in the closed subspace generated by E.

Note that we wrote strict inequality in (48) because we exclude the

countable set of values [alpha](p, A) where the maximum of [g.sub.p, A] (B) = [alpha]B - [f.sub.1] (B) is not unique.

The solutions, oscillating with the cut-off frequency, appear to be arranged in a new

countable set of the fields generated by Eqs.

These perspectives either have been, or could be, used to describe thinking about

countable sets (in our case, the set of natural numbers N).

Another curiosity is that in both places, the theorem as stated - 'ZF plus V = L has an [Omega]-model which contains any given

countable set of real numbers' - appears to be [[Sigma].sub.2], whereas what is proven is the [[Pi].sub.2] sentence that every

countable set of reals s is representable in some constructible [Omega]-model.

Theorem 2 Let X be a separable Banach space and let S be a

countable set in X.

X =

countable set; [[tau].sub.i] = cofinite topology and [[tau].sub.j] = discrete topology, is pairwise semi second countable.

if {[U.sub.m] : m [element of] N} is a basis of topology on C then [Mathematical Expression Omitted] is a basis of the topology on X.) Using Urysohn's lemma this can be chosen to be a

countable set if X is metrizable.

Given a

countable set A [subset] [C.sub.p](X) let [phi] : X [right arrow] [C.sub.p](A) be the reflection map.

Assume that K acts on some

countable set X, and take any nontrivial group Z.

From this and since [v.sub.[alpha]] is positive, we have that there exists a

countable set N [subset] [DELTA] such that

Note that, by separability of K, a locally convex space E over K is separable iff there is a

countable set whose linear hull is dense in E (iff the space is of countable type, in case E is metrizable).