A General Topology Workbook by Iain T. Adamson

By Iain T. Adamson

This booklet has been known as a Workbook to make it transparent from the beginning that it's not a standard textbook. traditional textbooks continue via 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 ebook will certainly locate all of the traditional constituents--definitions, theorems, proofs, examples and workouts­ yet now not within the traditional association. within the first a part of the e-book could be stumbled on a short 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 value and usability. ) The routines are intentionally now not "graded"-after the entire difficulties we meet in mathematical "real lifestyles" don't are available order of hassle; 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 rather tough effects. No strategies 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 attempt their hand at fixing the issues and proving the theorems for themselves.

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Certainly f->(A) = {O} and f->(B) = {I}. We have to show that f is continuous. The usual topology on [0, 1] is generated by the collection of subsets of the two types [0, s) and (t, 1]; so we need only show that the inverse images und er f of such sets are T-open. To do this we prove first that 1-[0, s ) =: U '·EJ(U,. )) where L is th e set of dyadic rationals in (t, 1]. Theorem 5 = Exercise 168 (Tietze's Extension Theorem) . Let (E, T) be a normal topological space, FaT-closed subset of E . Then every continuous mapping from F to a bounded closed interval I of R can be extended to a continuous mapping from E to I .

To prove the only if part, let (E , T) be a completely regular space. Let F be th e set of all cont inuous mappings from E to [0, IJ; for each f in F let If = [0,1] a nd let P = n f EF If . Define a mapping 9 from E to P by set t ing g(x) = (J (x ))fEF for all x in E . Prove first (using the T) property) that 9 is injective (so th at we have an inverse mapping g-I from g~(E) to E) ; th en that g is cont inuous; then finally th at g-I is continuous. (7) The topology T and th e topologi cal space (E ,T) are said to be normal if, for every pair of disjoint T-closed subsets A a nd B of E , there exist disjoint T-open subsets U a nd V including A and B resp ectively.

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. Let E = 0 iEI E, and, for each index i in J, let 1I"i be the projection mapping from E to E i . The topology induced on E by the family of mappings (1I"i)iEI is called the product topology on E ; we denote it by 0 iEI i: The topological space (0 iEl e; 0 iElTi) is called the topological product of the family ((Ei ,Ti)) iEI.

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