Question

Problem 4. Determine if the following sets B1, B2, B3, B4 and Bs are open, closed, compact or connected. (You don't need to prove your findings here) a) B1 =RQ. b) We define the set B2 iteratively: C1 = [0, 1] C2 =[0,1/4] U [3/4, 1] C3 =[0,1/16] U [3/16, 4/16] U [12/16, 13/16] U [15/16, 1] Then B2 = n Cn. NEN c) B3 = U (2-7,3+"). nn +1 NEN d) f:R+R continuous and V CR closed. B4 = f-'(V). Remark: Only provide those properties which are necessarily true. You might not be able to decide on some of the properties. e) f:R → R continuous and bijective, i.e. onto and one-to-one. B; = f -([0, 1] U (2, 3]). Problem 5 Tot ho c onocando 1 cuch that for 11 N

Answer 1

@ B, = TRIQ = 9° = set of irrational number Hence B, is neither oper nor closed. B, is not compact since it is not closed B is not connected since every two irrational number there is a rational number. non nie not closed B2 = non where each Cn is closed NEN of closed set is closed we know intersection is closed. L Hence B2 is closed, B2 is compact since so Be is not open, e each en is compact - By is connected (c) By=U (2-5o, 3+ hati) htu NEN B = union of arbitrary open set hence Bz is open Bz is not closed since it is open By is connected since (2- , 3. horta o are connected with non empty intersection. By is not compact since it is not closed.

S By = ftve) ver is closed. since f is continous so inverse image of closed is closed. Hence Byo is a closed set t em a Bu is not open since it is not closed. since we not know that is compact or connected so it is not nesse necessary that ft() is compact or connected compact or Hence By is not necessary connected,