On Nets and Filters.

Notations and terms not defined here are as in [3]. Ex denotes either an example or an exercise. The choice is usually up to you the reader, depending on the amount of work you wish to do. Those which direct or expect you to verify something, however, should be done. Most of the first three sections is adapted from [2]. The proof at the end of Section 3 is taken from [3]. 1 Filters Definition 1 A filter is a non-empty collection F of subsets of a topological space X such that: 1. ∅ / ∈ F ; 2. if A ∈ F and B ⊇ A, then B ∈ F ; 3. if A ∈ F and B ∈ F , then A ∩ B ∈ F. A maximal filter is also called an ultrafilter. Ex The set of all neighborhoods of a point x ∈ X is a filter N x called the neighborhood filter of x. Definition 2 A filter F is said to converge to x ∈ X, denoted by F → x, if and only if every neighborhood of x belongs to F. Defining a subfilter in the obvious way, this is equivalent to saying that N x is a subfilter of F. Ex N x and U x both converge to x.