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4 Basic Definitions A I K VI a b Figure 1. The cylinder and the Mobius band Exercise 8. Prove that any continuous image of a compact space is com- pact. Proble m 4. Let K be a compact metric space with metric p. Suppose that / : K — K is a continuous map such that p(f(x),f(y)) p(x,y) for any distinct x,y G K. Prove that / has a fixed point. On the Cartesian product 1 x 7 of topological spaces X and Y, the product topology is defined. The open subsets of X x Y in this topology are the Cartesian products of open subsets of X and Y and their unions. The product topology arises from the natural requirement that the pro- jections px{x,y) — x and py(x,y) — y be continuous. Indeed, for these maps to be continuous, the sets U x Y and X x V must be open for every open U C X and V C Y. The minimal topology o n l x F containing all these sets coincides with the product topology. ' Note that the Cartesian product S1 x I, where I is the interval [0,1], is the cylinder (Figure la) rather than the Mobius band (Figure lb). The point is that, although the Mobius band admits a natural projection onto S1, is does not admit any natural projection onto / . For any subset Y of a topological space X , the quotient space X/Y is obtained by identifying all the points of Y. Thus, the points of the space X/Y are all the points of the set X \Y and one point Y. A subset of X/Y is open if and only if its preimage under the natural projection p: X — • X/Y is open. A quotient space can also be defined for a space X on which an equiv- alence relation ~ is given. The points of the quotient space X/ ~ are the equivalence classes a subset of Xj ~ is open if its preimage under the nat- ural projection p: X — X/ ~ is open. (Declaring that x\ ~ X2 if and only' if x\ = #2 or #i, X2 G Y C X , we obtain the preceding construction.)