By Joseph J. Rotman

**Publish yr note:** First released in 1988

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A transparent exposition, with routines, of the elemental rules of algebraic topology. compatible for a two-semester direction in the beginning graduate point, it assumes a data of aspect set topology and uncomplicated algebra. even though different types and functors are brought early within the textual content, over the top generality is refrained from, and the writer explains the geometric or analytic origins of summary strategies as they're brought.

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**Extra resources for An Introduction to Algebraic Topology (Graduate Texts in Mathematics, Volume 119)**

**Example text**

Show that X and Y do not have the same homotopy type. 5. 6. Contractible sets and hence convex sets are connected. 7. Let X be Sierpinski space: X = {x, y} with topology {X, 0, {x}}. Prove that X is contractible. 8. (i) Give an example of a continuous image of a contractible space that is not contractible. (ii) Show that a retract of a contractible space is contractible. 9. If J: X -+ Y is nullhomotopic and if g: Y homotopic. -> Z is continuous, then go J is null- The coming construction of a "cone" will show that every space can be imbedded in a contractible space.

3. 2) that if f and 9 are paths with 9 constant, then I' ~ g' reI {1} if and only if there is a free homotopy I' ~ g'. The Fundamental Groupoid 41 Definition. If I: I .... ). A path I in X is closed at Xo if CL(f) = Xo = w(f). ) = meg); therefore we may speak of the origin and end of a path class and write CL[fJ and w[f]. E X, then the constant function ip: I .... X with ip(t) = P for all tEl is called the constant path at p. If I: I .... X is a path, its inverse path 1-1: I .... X is defined by tl-+ 1(1 - t).

8 applies. Finally, if V is an open set in Y, then f-l(V) is an open set of the stated form: f-l(V) = f-1f(f-l(V)); the result now follows easily. D Remark. If A is a subset of X and h: X --+ Z is constant on A, then h is constant on the fibers of the natural map v: X --+ X/A. 10. Let X and Z be spaces, and let h: X --+ Z be an identification. Then the map