Question

Instruction: Do any 10 of the 14 questions. Each question is worth 10 points. (For each True/False question, if it is true, answer T and give reasons for your answer. If it is false, answer F and give a explicit counterexample or other explanation of why it is false.) 1) True or False: The set {x : x = : x = tany, y e [0,5)} is an compact subset of R, the set of all real numbers with the euclidean topology. 2) True or False: T = = {0, R} U{[a, 0): € R} is a topology on R. 3) True or False: Let T1 and T2 be two topologies on a set X. If T3 is defined by T3 = T U T2, then T3 is not necessarily a topology on X. 4) True or False: Let Ti and T2 be two topologies on a set X. Then Ti is said to be a finer topology than T2 if T1 T2. The Euclidean topology R is finer than the discrete topology on R.

Instruction: Do any 10 of the 14 questions. Each question is worth 10 points. (For each True/False question, if it is true, answer T and give reasons for your answer. If it is false, answer F and give a explicit counterexample or other explanation of why it is false.) 1) True or False: The set {x : x = : x = tany, y e [0,5)} is an compact subset of R, the set of all real numbers with the euclidean topology. 2) True or False: T = = {0, R} U{[a, 0): € R} is a topology on R. 3) True or False: Let T1 and T2 be two topologies on a set X. If T3 is defined by T3 = T U T2, then T3 is not necessarily a topology on X. 4) True or False: Let Ti and T2 be two topologies on a set X. Then Ti is said to be a finer topology than T2 if T1 T2. The Euclidean topology R is finer than the discrete topology on R.

Answer 1

SIN/ (1) False Explanahon let A= {xor 1 artany ; Y6 (0,512)} Since tan 3--)00 as Y-39 Coty 8-10 Hence in the obd af / H (04-6,5%) i t fy Un-bu uncled sne Hence by Hein. Borel thovem ACR is Not Compacho Note: H. B. theorem Closed & bounced sub-set af R is Compact won.t usual topology 0 (2) True le bea · Explanahon : 'T= {0, P2} v{ [q.co) lacp (right half open ray topology) because it is closed W.Vit finite intersechun & Qobrtvary union (a,0) D ( 5,00) ca, 0); [b,a ; acb [au, a) [ j then U ((a.)) = [an,a) et EN Lindexskt for if index set is infinite, then I (ait) IR D n= int (N) to B). False letx= {1,2,34 Counler : t = {4, {"},x4, t ={°,+24,8% = G = GIUG = {0,111,427,4} f140f24 = {1,214 [3 =) ( is Not topology (4) True E2h =) [, is finer & Q is coarsev tupolugy C by definahun) Scanned with CS CamScanner