6. Let (X,T) be a topological space such that every subset of X is closed, thena. (X,T) is an indiscrete space.b. (X,T) is a discrete spacec. T is the finite closed topology on Xd. None of the above7. Let X be an infinite set equipped with the finite closed topology. A finite subset of X isa. closedb. орenc. clopend. neither open nor closed8. We define the real valued function f(r) = r. The inverse image of {-9,-1,0, 4} isa. {0,1,2, 3}b. {0,2}c. {-2,0, 2}d. None of the abovex- 2, r<0,|r+2, r20'9. Let f : R +R be defined by f(x) =, then f-(]1,3[) (the inverse image of theinterval ]1, 3[ under f) is(a) ]3,5[U] – 1,1[(b) [0, 1[(c) 1-1,1[(d) None of the above10. Let (X, Tx) and (Y, Ty) be two topological spaces, such thatX = {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 byf(a) = f(b) = 2, f(c) = 4, f(d) = 3g(a) = g(b) = 9(d) = 2, g(c) = 3Then(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

By theorem, in a topological space X,T, a subset A of X is close if and only if its complement Ac =X…

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

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

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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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