By Solomon Lefschetz

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**Additional info for Algebraic Topology (Colloquium Pbns. Series, Vol 27) **

**Example text**

A space is said to be contractible if it is homotopy equivalent to the one-point space. 5. Proposition. A space X is contractible ¢> the identity map 1x: X -4 X is homotopic to a map r: X -4 X whose image is a single point. PROOF. Let Y = {xo} = im(r). Then we have the inclusion map i: Y -4 X and the retraction r: X -4 Y. Now roi = 1y and ior ~ 1x by assumption. The D converse is also easy. 6. Example. Consider euclidean space X = Rn and the homotopy F: X x I -4 X given by F(x, t) = tx. This is a homotopy between fl = 1x and fo, which is the map taking everything to {O}.

One can factor the projection of the disk to the sphere through this space. 9 one can show that the resulting map D nIsn - 1 -> sn is a homeomorphism. 9 is not available, the following gives a criterion for deciding the same sort of question. 11. Definition. If A c X and if ~ is an equivalence relation on X then the saturation of A is {xEXlx ~ a for some aEA}. 12. Proposition. If A c X and ~ is an equivalence relation on X such that every equivalence class intersects A nontrivially, then the induced map k: AI ~ -> X I ~ is a homeomorphism if the saturation of every open (resp.

Let X be a compact metric space and let {Va} be an open covering of X. Then there is a c5 > 0 (a "Lebesgue number" for the covering) such that (A eX, diam(A) < c5) => A c Va for some tX. PROOF. For each XEX there is an E(X) > 0 such that B 2 €(x)(x) C Va for some tX. Then X is covered by a finite number of the balls B€(x)(x), say for x = Xl"'" xn- Define c5 = min {E(x;)1 i = 1, ... , n}. Suppose diam(A) < c5 and pick a point aoEA. Then there is an index 1::; i::; n such that dist(a o, x;) < E(XJ If aEA, then dist(a, a o) < c5 ::; E(XJ By the triangle 0 inequality, dist(a,x;) < 2E(xJ Thus A c B 2 €(Xi)(X;) c Va for some tX.