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.
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.
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).