Get A General Topology Workbook PDF

By Iain Adamson

ISBN-10: 081763844X

ISBN-13: 9780817638443

This e-book has been known as a Workbook to make it transparent from the beginning that it isn't a traditional textbook. traditional textbooks continue by way of giving in each one part or bankruptcy first the definitions of the phrases for use, the options they're to paintings with, then a few theorems concerning those phrases (complete with proofs) and at last a few examples and routines 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 exercises­ yet now not within the traditional association. within the first a part of the publication may be stumbled on a brief evaluation of the fundamental definitions of basic 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 trouble yet of value and usability. ) The workouts are intentionally now not "graded"-after the entire difficulties we meet in mathematical "real life" 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 tough 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 attempt their hand at fixing the issues and proving the theorems for themselves.

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Sample text

The proof for the countable case follows by replacing finite by countable in the proof above. 7. A closed cell of a CW complex is a metrizable subset. Proof. Let be a closed cell of X and choose a characteristic map ",:Dn--t(J, where Dn has the standard metric d. Define a metric d on by the formula d(x,y) =d(",-I(X),,,,-I(y». Since ",-I(X) is compact in Dn, this is always well defined. That this is a metric is easy to verify, the most difficult point being d(x,y) =0 implies x=y. But since ",-I (X) and ",-l(y) are compact, if d(",-l(x), ",-l(y» =0, then they must intersect, and thus x=y.

N of X I

Suppose that X is a space, and for each hE A, there is a map fA: EA ~X of the boundary of the Euclidean cell EA into X. Let 8=U{EAI hEA} ande=U{EA I hEA} be the disjoint unions, and let F=UAfA:e~X be the union map. The adjunction space XUj, 8 is said to be obtained by attaching the cells EA to X. The maps fA: EA ~X are called the attaching maps for the cells EA of XU j, 8. The adjunction of a family of O-cells is just the disjoint union. 1. Let X be a Hausdorff space, and suppose Y = XU j, 8 is obtained by attaching the cells {EA I h E A} to X.

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A General Topology Workbook by Iain Adamson

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