By Iain T. Adamson
This ebook has been referred to as a Workbook to make it transparent from the beginning that it's not a standard textbook. traditional textbooks continue via giving in every one part or bankruptcy first the definitions of the phrases for use, the techniques they're to paintings with, then a few theorems concerning those phrases (complete with proofs) and eventually a few examples and routines to check the readers' realizing of the definitions and the theorems. Readers of this e-book will certainly locate all of the traditional constituents--definitions, theorems, proofs, examples and routines yet no longer within the traditional association. within the first a part of the ebook might be discovered a short evaluation of the fundamental definitions of basic topology interspersed with a wide num ber of workouts, a few of that are additionally defined as theorems. (The use of the be aware Theorem isn't really meant as a sign of hassle yet of value and usability. ) The workouts are intentionally now not "graded"-after the entire difficulties we meet in mathematical "real existence" don't are available in order of trouble; a few of them are extremely simple illustrative examples; others are within the nature of instructional difficulties for a conven tional path, whereas others are really tricky effects. No options of the routines, no proofs of the theorems are integrated within the first a part of the book-this is a Workbook and readers are invited to attempt their hand at fixing the issues and proving the theorems for themselves.
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If B is a base for an ultrafilter on E then f-(B) is a base for an ultrafil ter on E'. T he proof of th e first assertion is straightforward using t he crite rion of Theorem 1. To prove the second assert ion, suppose th at B is bas e for an ultrafilter F on E and let F' be th e filter on E' generated by f -(B) . Show that F' is an ultrafilter by using Theorem 4 (let X' be any set in F' and conside r th e two cases f -(X') E F and f -(X') ¢ F) . Theorem 7 = Exercise 109. Let f be a mapping from a set E t o a set E'.
Let (D ,:::;) be a directed set, E any set. A mapping v from D to E is called a net in E with domain D. 38 Chapter 4 Let v be a net in E with domain D, A a subset of E . We say that v is eventually in A if there exists an element k in D such that v(n) E A for all elements n in D such that n 2:: k . We say that v is frequently in A if for every element n of D there exists an element n ' of D such that n ' 2:: nand v( n') E A. Let D and D I be directed sets, v and Vi nets in E with domains D and D I respectively.
This observation allows us to construct a net associated with F which is not eventually in V and so does not converge to x. Theorem 14 = Exercise 124. Let (E ,T) be a topological space, v a net in E and x a point of E. Then v converges to x if and only if the filter F(v) converges to x. This follows at once from the definitions of the terms involved . Theorem 15 = Exercise 125. Let (E, T), (E', T') be topological spaces, f a mapping from E to E' and x a point of E. Then f is (T, T')continuous at x if and only if for every net v in E which converges to 40 x the net f Chapter 4 0 v in E ' converges to f(x) .