### Define an equivalence relation on P such that 12/ is the Mobius band....

Short Answers and ProlL Choose 5 (10 pts each) 7"1.-.± T' P PI—n- . a p ifi 7. Prove: P2#P2 V K B ‘L. B ove that (0, co) is homeomorphic to (—o0,00). 0+64 i,5.; 4 Set X = {a, bit, d) t/ (a) Which of the following sets define a topology on X ? .0-r (i) (0. {a, 6) , (c,d), 0, c, d}, X) MO t TO P4 21°97' (IV C r' (ii) (0, (a), (c) , (a , c) , {b. c}, {a, bin} , (6, c, d), X), p/01 os 'gyp .4/C •I ('3 ii (iii) (0, (b), (c),(iticijacjii (a , c, d) , (b,c,d), X) Hot o lo p 0 1 lot C ci

Give a reason for any which do not. (b) Let 0 be the topology ,-‘4, Slit (0 , .0), {a, b}, (b, d), {a, 6,4}, {b, c, d} , X) on X. Is the set c, d closed in (X, 0)? spy A 9.0,-.1 -T Give a reason. ek../. I) icy-pwe,n ' .becr e ... V S-( Se o .a I- t 5-p.-• n 1.0. (a) What surface do we obtain lrom the identification polYTOn corresponding to the word aba-lb? MoVio.1 51—vic' (b) What surface do we obtain from the identification polygon corresponding to the word abcdectelbarl Kr n lab-\4-1-12 . • ..,.._ et. (t) Find all distinct identification polygons with two or four sides. (b) Write the word associated to the identification polygons. • f2PIdentify the corresponding surfaces.. •• d) Describe a method to decide whether the surface represented by a polygon is orieatable or not by looking at the polygon. SState the classification theorem of closed surfaces.

X(a) Define an equivalence relation on P such that 12/ is the Mobius band. (b) Define an equivalence relation on 12 such that P/ is 7'2.

Megha
27-Apr-2020