New PDF release: A General Topology Workbook

By Iain T. Adamson

ISBN-10: 081763844X

ISBN-13: 9780817638443

ISBN-10: 0817681264

ISBN-13: 9780817681265

This ebook has been referred to as a Workbook to make it transparent from the beginning that it isn't a standard textbook. traditional textbooks continue by means of giving in every one part or bankruptcy first the definitions of the phrases for use, the thoughts 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' realizing of the definitions and the theorems. Readers of this e-book 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 might be came upon a short overview of the elemental definitions of common topology interspersed with a wide num­ ber of routines, a few of that are additionally defined as theorems. (The use of the note Theorem isn't meant as a sign of hassle yet of significance and usability. ) The workouts are intentionally no longer "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 instructional difficulties for a conven­ tional direction, whereas others are relatively tricky effects. No strategies of the routines, 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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Extra info for A General Topology Workbook

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28 Chapter 3 Example. Let (E ,T) be a topological space, R an equivalence relation on E , 7] th e canonical surj ection from E onto E / R (given by 7](x) = R-class of x for each element x of E) . The topology coinduced on E / R by the famil y consist ing of the singl e mapping 7] is called the quotient topology on E / R; it is denoted by T / R . We call (E/ R, T / R) th e quotient space of (E, T) with resp ect to R. Let (E ,T) , (E' ,T') be topologi cal spaces, f a (T, T')-continuous mapping from E to E' .

Let (E, T) be a topological space. Let A be the relation defined on E by setting (x, y) E A if and only if x E CI T{Y}' Exercise 133. Show that the relation A is reflexive and transitive. Show also that A is antisymmetric if and only if the topology T is To. A topological space (E, T) is called an Alexandrov space if the intersection of every family of T-open subsets of E is T-open. Exercise 134. Let (E, T) be an Alexandrov space. Show that a subset K of E is T-closed if and only if whenever y E K and (x, y) E A (where A is the relation defined before Exercise 133) we have x E K.

Exercise 89 follows from th e definition of th e quotient topology T / R J and the comments we hav e just made about saturated sets. Both dir ections of Exercise 90 are straightforward. Exercise 91 is best handled by proving th e three implications (1) ===} (2) , (2) ===} (3) , (3) ===} (1). Theorem 5 = Exercise 89. With th e notation just described, is a (T/R J , (T') [J)-homeomorphism if and only if, for every R r saturated T-open subse t U of E , we have f ~(U) E T D . r Let (E ,T) be a topological sp ace, R a n equivalence relation on E .

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A General Topology Workbook by Iain T. Adamson

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