By Iain T. Adamson

This ebook has been known as a Workbook to make it transparent from the beginning that it's not a traditional textbook. traditional textbooks continue through giving in each one part or bankruptcy first the definitions of the phrases for use, the ideas they're to paintings with, then a few theorems regarding those phrases (complete with proofs) and eventually a few examples and workouts to check the readers' knowing of the definitions and the theorems. Readers of this booklet 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 ebook should be discovered a brief evaluation 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 meant as a sign of trouble yet of significance and value. ) The routines are intentionally now not "graded"-after the entire 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 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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**Example text**

Let (E, T) be a topological space, A and B subsets of E such that E = AU B . Let M be a subset of A n B which is both TA-open and Tn-open. Prove that M is T-open. Exercise 80. Let (E ,T) be a separable topological space. Prove that if V is a T-open subset of E then (V ,Tv) is also separable. Exercise 81. Let E be an uncountable set, p a point of E and Tp the particular point topology on E determined by p. Let A = Cdp}. Prove that (E, T p ) is separable but that (A, (Tp)A) is not. Example 2. Let ((Ei ,Ti))iEI be a family of to pological spaces.

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. Exercise 165. Show that the digit al top ology on Z is not normal. Exercise 166. Let (E, T) be a topological space. Show th at (E ,T) 53 Separation Axioms is normal if and only if, for every T-open set U and every T-closed subset A of U , there is a T-open set U' such that A ~ U' and CI U' ~ U . Theorem 4 = Exercise 167 (Urysohn's Lemma) . Let (E, T) be a normal topological space.

Space is oft en called a Tihonov space. 2 Separation Axioms 51 As we remarked in Section (5) the terminology in the literature is not entirely consistent: some authors interchange the meanings of completely regular and T 3 1 . 2 There is an example of a space, called the Tihonov corkscrew, which is T3 but not T3 1 . ] Exercise 161. Let E = {(x, y) E R2 : y :2: O} . For each point (x, y) of E such that y > 0, let B(x, y) be the collection of sets of the form E n V((x, y), c) for all positive real numbers e ; for each point (x,O) of E, let B(x, 0) be the collection of sets of the form V( (x, c), c) U {(x, for all positive real numbers e ; let B = U (x,y)EE B(x, y).