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1. Let X {1,2, 3, 4}. A topology on X is a. T = {6, X, {1}, {2}} b. T = {6, X, {1}, (3, 4}} c. T3 = {6, X, {1}, {1,3, 4}} d. None of the above

1. Let X {1,2, 3, 4}. A topology on X is a. T = {6, X, {1}, {2}} b. T = {6, X, {1}, (3, 4}} c. T3 = {6, X, {1}, {1,3, 4}} d. None of the above

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter4: Quadrilaterals
Section4.1: Properties Of A Parallelogram
Problem 44E
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Choose the correct answer and attach the details of your work. 1. Let X = {1,2,3, 4}. A topology on X is a. T = {0, X, {1}, {2}} b. T2 = {6, X, {1}, {3, 4}} c. T3 {6, X, {1}, {1,3, 4}} d. None of the above 2. Let X = {1,2, 3, 4} and let T = {6, X, {1}, {3}, {1,3}} be a topology on X, then (2, 4} is %3D a. open in (X, T) b. closed in (X, T) c. clopen in (X,T) d. neither open nor closed 3. Let (X,T) be the topological space defined in question 2., then the closure of (2} is а. X b. {2, 4} c. {1,2, 4} d. None of the above 4. Let (X,T) be the topological space defined in question 2. Let A = {1,2,3}, then the set of limit points of A is a. A b. {4} c. {2, 4} d. None of the above 5. Let (X,T) be a discrete space, then Va e X, the set {a} is а. орen b. closed c. clopen d. is neither open nor closed
Transcribed Image Text:Choose the correct answer and attach the details of your work. 1. Let X = {1,2,3, 4}. A topology on X is a. T = {0, X, {1}, {2}} b. T2 = {6, X, {1}, {3, 4}} c. T3 {6, X, {1}, {1,3, 4}} d. None of the above 2. Let X = {1,2, 3, 4} and let T = {6, X, {1}, {3}, {1,3}} be a topology on X, then (2, 4} is %3D a. open in (X, T) b. closed in (X, T) c. clopen in (X,T) d. neither open nor closed 3. Let (X,T) be the topological space defined in question 2., then the closure of (2} is а. X b. {2, 4} c. {1,2, 4} d. None of the above 4. Let (X,T) be the topological space defined in question 2. Let A = {1,2,3}, then the set of limit points of A is a. A b. {4} c. {2, 4} d. None of the above 5. Let (X,T) be a discrete space, then Va e X, the set {a} is а. орen b. closed c. clopen d. is neither open nor closed
6. Let (X,T) be a topological space such that every subset of X is closed, then a. (X,T) is an indiscrete space. b. (X,T) is a discrete space c. T is the finite closed topology on X d. None of the above 7. Let X be an infinite set equipped with the finite closed topology. A finite subset of X is a. closed b. орen c. clopen d. neither open nor closed 8. We define the real valued function f(r) = r. The inverse image of {-9,-1,0, 4} is a. {0,1,2, 3} b. {0,2} c. {-2,0, 2} d. None of the above x- 2, r<0, |r+2, r20' 9. Let f : R +R be defined by f(x) = , then f-(]1,3[) (the inverse image of the interval ]1, 3[ under f) is (a) ]3,5[U] – 1,1[ (b) [0, 1[ (c) 1-1,1[ (d) None of the above 10. Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b,c, d}, Tx = {0, X, {a}, {a, b}, {a, b, c}} Y = {1,2,3, 4}, Ty = {ó,Y, {2}, {2,3, 4}} Consider the functions f : X →Y and g: X Y defined by f(a) = f(b) = 2, f(c) = 4, f(d) = 3 g(a) = g(b) = 9(d) = 2, g(c) = 3 Then (a) f and g are both continuous (b) f and g are both discontinuous (c) f is discontinuous and g is continuous (d) f is continuous and g is discontinuous
Transcribed Image Text:6. Let (X,T) be a topological space such that every subset of X is closed, then a. (X,T) is an indiscrete space. b. (X,T) is a discrete space c. T is the finite closed topology on X d. None of the above 7. Let X be an infinite set equipped with the finite closed topology. A finite subset of X is a. closed b. орen c. clopen d. neither open nor closed 8. We define the real valued function f(r) = r. The inverse image of {-9,-1,0, 4} is a. {0,1,2, 3} b. {0,2} c. {-2,0, 2} d. None of the above x- 2, r<0, |r+2, r20' 9. Let f : R +R be defined by f(x) = , then f-(]1,3[) (the inverse image of the interval ]1, 3[ under f) is (a) ]3,5[U] – 1,1[ (b) [0, 1[ (c) 1-1,1[ (d) None of the above 10. Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b,c, d}, Tx = {0, X, {a}, {a, b}, {a, b, c}} Y = {1,2,3, 4}, Ty = {ó,Y, {2}, {2,3, 4}} Consider the functions f : X →Y and g: X Y defined by f(a) = f(b) = 2, f(c) = 4, f(d) = 3 g(a) = g(b) = 9(d) = 2, g(c) = 3 Then (a) f and g are both continuous (b) f and g are both discontinuous (c) f is discontinuous and g is continuous (d) f is continuous and g is discontinuous
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