By Iain T. Adamson

This publication has been known as a Workbook to make it transparent from the beginning that it's not a traditional textbook. traditional textbooks continue by means of giving in each one part or bankruptcy first the definitions of the phrases for use, the suggestions they're to paintings with, then a few theorems related to those phrases (complete with proofs) and at last a few examples and workouts to check the readers' knowing of the definitions and the theorems. Readers of this ebook will certainly locate the entire traditional constituents--definitions, theorems, proofs, examples and routines yet no longer within the traditional association. within the first a part of the booklet might be came across a brief evaluate of the elemental definitions of normal topology interspersed with a wide num ber of workouts, a few of that are additionally defined as theorems. (The use of the note Theorem isn't meant as a sign of trouble yet of significance and usability. ) The routines are intentionally no longer "graded"-after all of the difficulties we meet in mathematical "real lifestyles" don't are available in order of hassle; a few of them are extremely simple illustrative examples; others are within the nature of instructional difficulties for a conven tional direction, whereas others are rather tricky effects. No ideas 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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Let A be a collection of subsets of a non- empty set E which generates a filter F on E . If for every subset X of E we have eithe r X E A or GdX) E A , then A is an ultrafilter on E. To prove Theorem 4, suppose F' is an ultrafilter which includes F and show that if X is a ny set in F' then GB(X ) (j. A an d hence X E A. Corollary. 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.

T a nd (E ,T) ar e sa id to be T 4 if t hey ar e both T 1 and normal. Here aga in th ere is some confusion over terminology: some aut hors use T 4 where we have used n ormal and normal where we have used T 4 . It can be shown that the Nemytskii space is Tal but not T4 . ] Let Wo be the first infinite ord ina l, W I th e first uncountabl e ordinal , 0 0 = [0, wo], 0 1 = [0, wd with th e ord er to pology in each case . Then T P = 0 0 x 0\ equipped wit h th e product topology is called th e Tihonov plank; it ca n be shown to be a T" space.

Prove that the digital topology is To but not T!. Separation Axioms 45 Exercise 137. Show that the topology T q induced by a quasimetric on a set E is T 1 • Theorem 1 = Exercise 138. Let (E ,T) be a topological space. The following conditions are equivalent: (1) T is a T 1 topology; (2) For every point x of E the set {x} is T -closed ; (3) For every point x of E th e intersection of th e T-neighbourhood filter of x is {x} . Prove the implications in the order (1) ==? (2) , (2) ==? (3), (3) ==? (1) ; all three are straightforward.