By Iain T. Adamson
This publication 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 each one part or bankruptcy first the definitions of the phrases for use, the innovations 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 booklet will certainly locate the entire traditional constituents--definitions, theorems, proofs, examples and routines yet now not within the traditional association. within the first a part of the e-book may be stumbled on a brief evaluation of the fundamental definitions of common 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 significance and value. ) The workouts are intentionally no longer "graded"-after all of the difficulties we meet in mathematical "real existence" don't are available order of hassle; a few of them are extremely simple illustrative examples; others are within the nature of educational difficulties for a conven tional path, whereas others are relatively tough effects. No suggestions of the workouts, 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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The continuity of j gives a T-open subset U of E containing x such that j- (U) ~ U I. Show that U x V' does not meet G. Exercise 152. Show that if E is a finite set then the only Hausdorff topology on E is the discrete topology. This follows easily from Exercise 140. Alternatively we may apply the Hausdorff property to all the pairs (Xl, X2), . , (Xl, X n ) to obtain open sets U2 , •• • , Un containing Xl but not X2, .. , X n; then consider the intersection of these open sets. (4) The topology T and the space (E, T) are said to be T 2 1 or "2 completely Hausdorff if, for every pair of distinct points x, y of E, there exist T-neighbourhoods V, W of x, y respectively such that CI V n Cl W = 0.
E. if there is an element m of E such that (x , m) E R for all x in X and (m,m') E R for every element m' of E such that (x, m') E R for all x in X. e. an element a of E such that there is no element Con vergence 33 x of E for which (a, x ) E R except a its elf. Zorn 's Lemm a is equivalent to the Axiom of Choice which states th at the product of every famil y of non-empty sets is non- empty. In this book we accep t the validi ty of t he Axiom of Choice and so may make use of Zorn's Lemma. Theorem 2 = Exercise 102.
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.