lustmetions: You must work on this exam without assistance from otheis, mut ito not use mum& sources, except your textbook and class notes. All run answers must be justified. If you cannot prove something. you may earn partial ere¦Att tw noting what must be proved.
Do not write more than necessary. A perfect answer is both cartrrt and oonerse
(6 '2) him, Theorem 6 2: A topological space X is connected if and only tf chute are no nonempty proper subsets of X that are both open and closed m
2. (0.r)2) Show that if X and Y are both path-conne•ted spaces. t hen 1s is oath connected 3, (a) (7 36) Let be a closed subset of a limit point compact space Show that in the sunspace topology A is limit point compact. (b) Show that la, b] n Q is not compact for all real numbers a • b .1. Find two non-compact spaces X and Y that arc not homeomorphic bat which have homeomorphic one-point compactifications. Justify both hut Suponse X is compact, V' is Ilausdorff. and f :.X --+ Y is a laject ivy oral, Show that f is a honwomorphism. I. Itecall the Cantor set. obtained by deleting middle thirds:
C E I E {0.2} \.,1 .-1
Show that C x C C, and in fact, the product of countably many copies of C is homcomorphic to C. Hint: Prove and use the following homeomorphism. If Xk is a wo_poinl discrete space for all k. then FM., X. C by f EIk E {0,2) Vk