Download e-book for iPad: Algebraic Topology: A First Course (Graduate Texts in by William Fulton

By William Fulton

ISBN-10: 0387943277

ISBN-13: 9780387943275

This ebook introduces the real principles of algebraic topology by way of emphasizing the relation of those principles with different parts of arithmetic. instead of picking one viewpoint of contemporary topology (homotropy thought, axiomatic homology, or differential topology, say) the writer concentrates on concrete difficulties in areas with a number of dimensions, introducing basically as a lot algebraic equipment as priceless for the issues encountered. This makes it attainable to determine a greater variety of vital positive factors within the topic than is usual in introductory texts; it's also in concord with the ancient improvement of the topic. The e-book is geared toward scholars who don't unavoidably intend on focusing on algebraic topology.

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Additional info for Algebraic Topology: A First Course (Graduate Texts in Mathematics, Volume 153)

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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Algebraic Topology: A First Course (Graduate Texts in Mathematics, Volume 153) by William Fulton

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