EXERCISES FOR MATHEMATICS 205C SPRING 2011 File Number 00 These are mainly review from the previous two courses in the sequence 1 i Let U be an open subset in Rn for some n Prove that U has countably many arc components Hint Why is every point in the same arc component as a point with rational coordinates ii Given two homotopy equivalent spaces X and Y prove that there is a 1 1 correspondence between their sets of arc components iii Let X be the Cantor Set constructed as n Xn where X0 0 1 Xn is a union of 2n pairwise disjoint closed intervals of length 3 n and Xn 1 is obtained from Xn by removing the open middle third from each interval Prove that X cannot have the homotopy type of an open subset in Rn for any n Hint Given two points u 6 v X find U and V be disjoint open subsets containing u and v such that U V X Why does this imply that every arc component of X consists of a single point 2 Let Y be a nonempty topological space with the indiscrete topology i e and Y are the only open sets and let X be an arbitrary nonempty topological space Prove that X Y consists of a single point Hint For all topological spaces W every map of sets from W to Y is continuous Using this show that if A B is a subspace and g A Y is continuous then g extends to a continuous map from B to Y 3 Let U be an open subset in Rn for some n and let u0 U be a point with rational coefficients i Let K be a compact metric space and let f K R n be continuous Prove that there is some 0 such that if the distance from x R n to f K is less than then x U Definition Let U be an open subset of R n A broken line curve is a continuous curve a b U such that the following holds There is a partition of a b given by a x 0 x1 x k b such that the restriction of to each closed subinterval x i 1 xi is a straight line segment which has a parametrization of the form xi 1 t t xi t xi 1 xi i i where i xi xi 1 The points a and b are called the initial and final points and the remaining points of the form x i are called corner points 1 ii Let 0 1 U be a closed curve such that 0 1 u 0 Prove that the class of is also represented by a broken line curve as above such that the initial and final points are 0 1 and the corner points i U have rational coefficients Hint Let 0 be as in the first point of the exercise use uniform continuity to find some 0 such that t s implies t s 41 and partition 0 1 into subintervals of length less than Suppose the partition is given by 0 x0 x1 xk 1 and for i 1 k 1 choose i such that i xi 41 and show that the corresponding broken line is base point preservingly homotopic in U to the original curve More precisely show that the line segment joining i to i 1 lies in the open disk centered at xi we know that the latter is contained in U iii Using the preceding explain why 1 U u0 is countable Hint finite sequences of points in U with rational coordinates Count the number of 4 Prove the compact support property of the fundamental group If 1 X x then there is a compact subset K of X such that x K and lies in the image of the map 1 K x 1 X x Furthermore if 0 and 0 in 1 K x map to the same element of 1 X x then there is some compact set L X such that L contains K and 0 and 0 map to the same element of 1 L x Hint If we choose representative curves or homotopies their images are compact 5 Let p E X be a covering map and let f Y X be continuous Define the pullback Y X E e y Y E f y p e Let p Y f projY Y X E i Prove that p Y f is a covering map Also prove that f lifts to E if and only if there is a map s Y Y X E such that p Y f s 1Y ii Suppose also that f is the inclusion of a subspace Prove that there is a homeomorphism h Y X E p 1 Y such that p o h p Y f NOTATION If the condition in ii holds we sometimes denote the covering space over Y by E Y in words E restricted to Y 6 i Suppose that A X is a retract and X is Hausdorff Prove that A is a closed subset of X Hint Let i A X be the inclusion and let r X A be the retraction What can we say about the set of all points y X such that i o r y y ii Suppose we have A B X such that A is a retract of B and B is a retract of X Prove that A is a retract of X iii Suppose that A is a retract of X let j A X be the inclusion mapping and let x 0 A If H is the image of the fundamental group of A under the mapping h prove that there is a normal subgroup K of 1 X x0 such that the latter is generated by H and K and we have H K 1 Hint Let r X A be the associated retraction and consider the kernel of r 2