|
Showing 1 - 2 of
2 matches in All Departments
An ultrafilter is a truth-value assignment to the family of subsets
of a set, and a method of convergence to infinity. From the first
(logical) property arises its connection with two-valued logic and
model theory; from the second (convergence) property arises its
connection with topology and set theory. Both these descriptions of
an ultrafilter are connected with compactness. The model-theoretic
property finds its expression in the construction of the
ultraproduct and the compactness type of theorem of Los (implying
the compactness theorem of first-order logic); and the convergence
property leads to the process of completion by the adjunction of an
ideal element for every ultrafilter-i. e., to the Stone-Cech com
pactification process (implying the Tychonoff theorem on the
compact ness of products). Since these are two ways of describing
the same mathematical object, it is reasonable to expect that a
study of ultrafilters from these points of view will yield results
and methods which can be fruitfully crossbred. This unifying aspect
is indeed what we have attempted to emphasize in the present work."
A chain condition is a property, typically involving considerations
of cardinality, of the family of open subsets of a topological
space. (Sample questions: (a) How large a fmily of pairwise
disjoint open sets does the space admit? (b) From an uncountable
family of open sets, can one always extract an uncountable
subfamily with the finite intersection property. This monograph,
which is partly fresh research and partly expository (in the sense
that the authors co-ordinate and unify disparate results obtained
in several different countries over a period of several decades) is
devoted to the systematic use of infinitary combinatorial methods
in topology to obtain results concerning chain conditions. The
combinatorial tools developed by P. Erdos and the Hungarian school,
by Erdos and Rado in the 1960s and by the Soviet mathematician
Shanin in the 1940s, are adequate to handle many natural questions
concerning chain conditions in product spaces.
|
|
Email address subscribed successfully.
A activation email has been sent to you.
Please click the link in that email to activate your subscription.