English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let w = 0, 1,.... }, and suppose that for each i E wc, Cl is a compact Hausdorff space with weight < c. A filter over co defines a topology on JJiEJ Ci . We prove that the continuum hypothesis implies the existence of ultrafilters over w for which the corresponding product space on REj,, Cl is paracompact. Moreover, we show that every P-point in / w w is an ultrafilter with this property. Since box products appear as closed subspaces of ultrafilter products, our theorem extends results of Mary Ellen Rudin ( 1972) and Kenneth Kunen (1978). 0. INTRODUCTION AND THEOREMS History. The study of normality and paracompactness in box products was initiated by Mary Ellen Rudin ([R], 1972) Kenneth Kunen ([Ku], 1978) and E. K. van Douwen ([vD1 ], 1975). (See also the survey papers of van Douwen [vD2], and Scott Williams [W].) Box products appear as closed subspaces of ultrafilter products. The Tychonoff topology of pointwise convergence and the box topology are examples of topologies on Cartesian products defined by filters over the index set. A filter can be used to define a topology as follows: take as a base element each Cartesian product of open sets where the set of indices on which the projection is the entire coordinate space is a set belonging to the filter. Then the filter consisting of sets with a finite complement defines the Tychonoff topology, while the improper filter consisting of the entire power set defines the box topology. C. J. Knight introduced the concept of a filter topology in the original paper on box products ([Kn], 1964). In this paper we show that the Rudin-Kunen method for proving paracompactness in the box product generalizes to a certain class of ultrafilter products. The decomposition theory of E. Michael ([M], 1957) underlies this method. Received by the editors February 21, 1986 and, in revised form, September 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 54D18; Secondary 54B10, 54B15, 54D40, 54G10.

Paracompact product spaces defined by ultrafilters over the index set