Book contents
Appendix: filters
Published online by Cambridge University Press: 05 April 2013
Summary
In topology nowadays the value of the concept of filter is generally recognized. Since I have made essential use of the concept, in the main part of the text, I am including as an appendix a brief account of the necessary theory, although no doubt it will already be familiar to many of my readers.
Definition (A.1). A filter on a given set X is a non-empty family F of non-empty subsets of X such that
(i) each superset of a member of F is a member of F,
(ii) the intersection of a finite subfamily of F is a member of F.
In a set X the most immediately obvious filters are those which consist of all supersets of a given non-empty subset. Such filters have the property that the intersection of any family of members is also a member. This is always the case for finite sets but infinite sets contain filters which do not have this property. For example the cofinite subsets of an infinite set form a filter F0 such that the intersection of all members of F0 is empty. (A cofinite subset is the complement of a finite subset).
- Type
- Chapter
- Information
- Introduction to Uniform Spaces , pp. 122 - 129Publisher: Cambridge University PressPrint publication year: 1990