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 by way of giving in every one part or bankruptcy first the definitions of the phrases for use, the ideas they're to paintings with, then a few theorems concerning those phrases (complete with proofs) and at last a few examples and routines to check the readers' realizing of the definitions and the theorems. Readers of this booklet will certainly locate all of the traditional constituents--definitions, theorems, proofs, examples and workouts yet no longer within the traditional association. within the first a part of the publication should be stumbled on a brief assessment of the elemental definitions of normal topology interspersed with a wide num ber of routines, a few of that are additionally defined as theorems. (The use of the observe Theorem isn't really meant as a sign of hassle yet of significance and value. ) The workouts are intentionally no longer "graded"-after all of the difficulties we meet in mathematical "real lifestyles" don't are available in order of trouble; a few of them are extremely simple illustrative examples; others are within the nature of educational difficulties for a conven tional direction, whereas others are relatively tough effects. No ideas of the workouts, no proofs of the theorems are incorporated within the first a part of the book-this is a Workbook and readers are invited to aim their hand at fixing the issues and proving the theorems for themselves.

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**Example text**

To prove the second part, let t/ be any net associated with an ultrafilter F. We must show that for every subset Y of E we have eit her (1) t/ is eventually in Y or (2) v is eventually in CE(Y) . Suppose (1) does not hold; then v is certainly frequ ently in CE(Y). We can show that F U {GE(Y)} generates a filter including F , which must therefore be F since F is an ultrafilter. So GE(Y) E F. e. v is eventually in GE(Y) . Chapter 5 SEPARATION AXIOMS Let (E, T) be a topological space. e. either a T -neighbour hood of x which does not contain y or a T-neighbourhood of y which do es not cont ain x.

Let (E ,T) , (E' ,T') be topologi cal spaces, f a (T, T')-continuous mapping from E to E' . Let R f be the equivalence relation on E defined by setting (x , y) E R f if and only if f (x) = f(y) ; let 7] be the canonical surjection from E onto E / R] . Let be th e ca nonical bijection from E/ R f onto B = f~(E) and j th e canonical injection from B to E' , so that f = j 0 0 T/ . Then T/ is (T, T / R f )-continuous and j is ((T') B, T')cont inuous. Furthermore, since 0 7/ is (T, (T') [J )-continuous, it follows from Theorem 4 th at is (T/ Rf , (T ' )lJ)-co nt inuous.

Let E be a non-empty set, a any element of E. Then the filter consist ing of all subsets of E which con tain a is an ultrafilt er on E . 34 Chapter 4 Exercise 105. Let F be an ultrafilter on a set E. Show that n F contains at most one point and that if F = {a} , then F is the ultrafilter consisting of all th e subset s of E which contain a. ) Exercise 106. Let A be a subset of a set E ; let F be a filter on E . Let FA be the set of subsets of A of th e form A n X where X is in F. Show that FA is a filter on A if and only if all t hese sets are non-empty.