Thus, [iota] will be injective on [mathematical expression not reproducible] for all but countably
many of the partitions P compatible with [[omega].sub.0].
In order to prove that F is a countably
strict-set contraction, we first consider
A subset D of L is countably
directed if every countable subset of D has an upper bound in D.
We remark that the functions in [V.sup.p] have lateral limits everywhere, and at most countably
A subset of X is I-sequentially compact if and only if it is I-sequentially countably
Using this and the definition of countably
additive, we see that [zeta] [member of] [[SIGMA].sub.m] [PHI]([A.sub.m]) iff for all [epsilon] > 0, there exists some N [member of] N and points [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [m.sub.i] < N so that [absolute value of [[SIGMA].sub.i][y.sub.i] - [zeta]] < [epsilon].
When dealing with such problems, we are given a finite or countably
infinite set of solutions from which we have to find the one that minimizes or maximizes the given function (Michiels et al.
For example, one can imagine such a "book" as being made up of a countably
infinite collection of six-by-nine inch "leaves" lying parallel to the xy-plane and projecting rigidly from each rational point in the unit interval of the z-axis.
184), while, in every formal system, only countably
many are available.
normed spaces on polyhedral domains, Proc.
(1970): "Markets with Countably
Many Commodities" International Economic Review, 11(3), 369-377.
Any infinite set with cardinal number [[omega].sub.1] can be expressed as a disjoint union of a class C of countably
infinite sets, the cardinality of C being [[omega].sub.1].