MATH 214 Final, Fall 2003

MATH 214: FINAL EXAM, JANUARY 28TH 2004 One piece of advice: half of the exam checks if you are familiar with basic definitions and examples. Assuming you studied for the exam, this should be very easy. I suggest you go through the exam and answer these easy questions first. If you feel something is not stated clearly, come over and ask me. But please - one by onel!! » Problem #1: --------- (20 points) rahlam 9- M15 nginte) +P 4 JO5 points * Problem #3: -_------- (25 points) * Problem #4: ---------. (35 points) * Problem #5: _-_----_ (25 points) » Problem #6: ---------- (30 points) * Total:___________________ (150 points) { “ibiagy OF etiAagT Problem #1 Topological Spaces / Bases / Subbases. (1) (5 points) Let (X,T)} be a topological space. and let B C 7. What does it mean when we say 8 is a basis for T? (2) (5 points) Let (X,7) be a topological space. and let S C JT. What does it mean.when we say S is a subbasis for 7? (3) (£0 points) Let S be a collection of subsets of X. Describe the topology Ts on X for which S is a subbasis. Then prove 7s indeed is a topology. Problem #2 Continuous Maps / Sequences. (1) (8 points) Let f : (X,T) > (¥,S) be a map between topological spaces. Define what it means for f to be continuous (2) (5 points) Let (a,) be a sequence of elements of a topological space (X.T) and let a € X. Define what it means an — @ in (X,T). (3) (5 points) Let (an) be a sequence of elements of (X,7), let a,b € X and let a, — a and a, — b. Isit possible that a 4 5? Justify your claim. m a, ae mete naa huerican BNIVERSIT IS i LIBRARY ( OF HIKE T -—-—___! Problem #3 Quotient Spaces / Product Spaces. (1) (5 points) Let f : (X.T) - (Y,S) be a map between topological spaces. Define what it means for f to be a quotient map. (2) (5 points) Give three examples of quotient maps. NO PROOFS please. (3) (5 points) Let {(X,.T,)|A € A} be a family of (non-empty) topological spaces. Define the Tychonoff topology on the product Thea X>». AMERICAN tare “iat (4) (10 points) Prove the projection maps Pra I XX 6 EA are quotient maps. If you find this hard, you may consider a simpler case when A = {1,2} - for half the credit. Problem #4 Separation Axioms. (1) (5 points) Define what it means for a topological space (X,7) to be: i) To ii) Ty iii) Tp (2) (5 points) Give an example of a space which is NOT T). Justify BRIEFLY. (3) (5 points). Define: i) regular spaces ii) normal spaces 7 (4) (10 points) Let (X,T) be NORMAL and let U,V € T satisfy VCOCKV) CU. Prove there exists W € T such that: VooCuvycw ccuw) cu. (5) (10 points) Let (X,7) be NORMAL and let U,V € T satisfy VCCUV) CH. Prove there exists f : X — R such that i) fly =l1 ii) Cl({x|f(e) 4 0}) CU Feel free to use part (4) even if you haven't solved it. 8 Problem #5 Compactness. (1) (5 points) Define what it means for a topological space (X, 7) to be: i} compact ii) locally compact iii) Lindeléf. (2} (5 points) Let (X,T) be a topological space. Define its one point compact- ification (X*,7*). § (3) G# points) Prove that the topological space (X*, 7”) from part (2) is com- pact. SOE iB AR. t or SADT . * ete canw ae, a 9 (4) (10 points} Prove that the topological space (X*,7*) from part (2) is Haus- dorff if and only if i) (X,T) is Hausdorff and ii) (X,T) is locally compact. 10 Problem #6 Connected ness. (1) (8 potnts) Define what it means for a topological space (X, T) to be: i) connected ji) locally connected iii) path connected iv) locally path connected (2) (10 points} Define (ONLY DEFINE!) the equivalence relations ~c and ~PC on X such that: i) the equivalence classes of of X; ~c are what we call connected components li) the equivalence classes of ~Pc are what we call path connected com- ponents of X. (3) (5 points) Prove that the relation ~pc Teally is an equivalence relation,