Filters in Analysis and Topology EFnetMath（在分析和拓扑EFnetMath过滤器）.pdfVIP

Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into metric spaces with subjects like completeness finding an elegant representation in terms of filters. The purpose of this paper is to provide a brief discussion of this general theory of filters, and attempt to demonstrate why they are an interesting and useful way to talk about convergence. It is not meant to be a complete introduction to the theory, but merely to provide enough background in it to give the reader a good idea of the essentials. 1 1 Filters and Convergence Let X be a topological space, and consider a point x ∈ X . Recall that we define a set V to be a neighbourhood of x if there is an open set U such that x ∈ U ⊆ V . Let Nx be the set of all neighbourhoods of a x. It is trivial to verify the following properties: 1. X ∈ Nx 2. If V ∈ Nx and V ⊆ W then W ∈ Nx 3. If U, V ∈ Nx then U ∩ V ∈ Nx 4. ∅ ∈ Nx If F is a collection of subsets of X which satisfy the above properties then we call it a filter . Nx is called the neighbourhood filter of x. Note that filters are closed under finite intersection as well as pairwise (by induction). An important example of a filter is the cofinite filter. Let X be an infinite set; then F = {U ⊆ X : X \ U is finite } Before we start discussing filters and convergence, we will want to prove and define various things about filters. To start with, lets consider ways of generating filters. It is an easy check that if C is a collection of filters on X then C is also a filter on X . So if there is a filter containing a collection of subsets of X , G the