MATH 214 Final, Fall 2018

Topology I - Math 214 Final Dr. Richard Aoun - Fall 2018 Fall 2018 Exercise 1. Endow R? with the Euclidean metric dz. For every R > 0, we denote by S(0, R) the sphere of center the origin and radius R and by B(0, R) the open ball of center the origin and radius R. *1. (a) (General question) Let X and Y be two topological spaces and f : X —> Y amap. Show that if X is connected and f is continuous, then f(X) is connected. (b) Using a suitable map f : [1, +00) x [0,27] —> R?, show that B(0,1)° := R? \ B(0,1) is connected. One can similarly prove (or deduce) that for every R > 0, B(0, R)° := R? \ B(O, R) is connected. You can use this fact later on without producing a proof. 2. (a) Let f : R? —> R be a continuous map. Let a € R. Show that if f7*({a}) is bounded, then there exists R > 0 such that f (B(O, R)©) € (-c0,a) or f (B(0,R)®) € (a, +00). (b) Let f : R? —> R be a continuous surjective map. Show that for every a € R, f-'({a}) is unbounded. Hint: For every R > 0, B(0, R) has compact closure. Exercise 2. A topological space X is said to be compactly generated if a subset A is closed in X whenever AN K is closed in K for every compact subset K C X. 1. (Preliminary) Let X be any topological space. Let (tn)nen be a sequence in X converging to some | € X. Show that K := {anjn > 0} U {I} is compact. 2. Deduce that any topological space that fulfills the conclusion of the sequence lemma (e.g. a metrizable space) is compactly generated. Exercise 3. We propose to find sufficient conditions for a continuous map between two topological spaces to be closed (recall that a map from a space X to a space Y is said to be closed if the direct image of any closed subset of X is closed in XS); el: Show that if X is a compact topological space and Y is a Hausdorff topological space, then any continuous map f : X — Y is closed. 2. A map f : X —>+ ¥ is said to be proper if for every compact subset K C Y, f~'(K) is compact. Cent Mrsurr (a) (Example) Show that ehiep f :R — Ris proper, if and only if, lim2|,400 |f(x)| = +00. (b) Let X be a topological space, Y a Hausdorff space and f : X —+ Y a continuous proper map. Let C be a closed subset of X. Show that for every compact subset K CY, f(C) NK is closed in Y. Hint: You can adapt some steps made in Question 1. Better, you can use it directly but in a suitable way. Remark and conclusion: Using the terminology of Exercise 2, we deduce that any contin- uous proper map f from a topological space X to a compactly generated Hausdorff space Y is a closed map. Exercise 4. Let (Kd) be a compact metric space and f : K —+ R a continuous map. 1. Show that f is uniformly continuous on K using two methods: (2) Method 1: using a suitable open cover of K. (b) Method 2: using the sequential characterization of compactness in metric spaces. Hint: Suppose that f is not uniformly continuous. Search then for two sequences (Tn )nen and (yn)een in K with a particular behavior. 2. Application: We take here K = [0,1!. We say that p: K —> R is piecewise-linear if there is a partition 0 = ag < a; < ... <a, = 1 of K such that p is linear on the interval {a,.@,41} for every 1 = 0,--- ,n. Let PL(K) < C(K) be the set of piecewise linear continuous functions on K. Show that PL(K) is dense in C(K) for the topology of uniform convergence. Bonus: Using Baire’s category theorem in a suitable complete metric space, show that there exists continuous functions on [0,1] which are nowhere monotone (i.e. not monotone on any subinterval of (0,1). Explain why Baire’s theorem implies actually that, in some sense to be precised rigorously, “almost all continuous functions on [0,1] are nowhere monotone”. Bonus bis*: Using Baire’s category theorem in a suitable complete metric space, show that there —° ” [0.1] which are nowhere differentiable (i.e. not differentiable at any x € (0, 1}). Explain why Baire’s theorem implies actually that, in some sense to be precised rigorously, “almost all continuous functions on {0,1} are nowhere differentiable”.