Exercise 1. Let X = {a, b, c, d, e}. Which of the following families is a topology?Why or why not?(1) 7= {0, {a}, {b}, {a, b}, {d, e}, X};T(2) T = {0, {a}, {b}, {a, b}, {d, c} , {a, b, c, d}, X};(3) T = {0, {a}, {b}, {a, b}, {d,c}, {a, b, c, d}};(4) T = {0, {a}, {a, b}, {b, c,d, e}, X};

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Answered: Exercise 1. Let X = {a, b, c, d, e}.…

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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Let X = {a, b, c}, T1 = {X, ̟, {a}, {b}, {a,b}} and T2 = {X, 6, {a}, {c}, {a,c}}. Then one of the following is true: the smallest topology on X containing both T1 and T2 is the discrete topology. O T1UT2 is a topology on X but TIOT2 is not O T1NT2 and T1UT2 are both topologies on X T1NT2 and T1UT2 both cannot be topologies on X
5. Show the span {(1,0, 1), (–1,2,3), (0, 1, – 1)} is all of R³. 6. Determine whether span {(1, –1,0), (0, 1, 1), (3, –5, -2)}, is all of R³.
Let X = {1, 2, 3, 4, 5, 6} and let T1 = {X, p, {3}, {1,2,3}, {2,3,4}, {2}} and T2 = {X, ¢, {2}, %3D {1,2,3}, {4,5,6}, {2,4,5,6}} * T1 is a topology on X and T2 is not a topology on X T1 and T2 are both not topologies on X O T1 and T2 are both topologies on X T1 is not a topology and T2 is a topology on X
2. Consider the topology 7, = {X,ø,{a},{b,c}}_on X = {a,b,c} and T; = {Y,. {u}} Y = {u,v} . the topology Note that on {(a,u),(a,v),(b,u),(b,v),(c,u),(c,v)} is the product set on which the product topology is defined. Determine the following: XxY = a) II.'(x) b) II,'(0) c) II.'(a) d) II,'(b,c) e) II, (u)
topology
1. Let S = {(1, 1), (1, 2), (2, 1), (3, 2), (3, 3)}. Consider the polytope conv(S). Write this set as {x | Ax 0}, where A has 3 rows.
2. Let X = {1,2, 3, 4} and let T = {ø, X, {1}, {3},{1,3}} be a topology on X, then {2,4} is a. open in (X, T) b. closed in (X,T) c. clopen in (X, T) d. neither open nor closed
Let X = {0, s, v}, Y = {c, d, e, i}, and Z = {0, 3, 6, 8, 9}. Let f : X → Y where ƒ f = {(o, c), (s, e), (v, d)}. Let g: Y→ Z where g = {(e, 9), (c, 8), (d, 3), (i, 6)}. g•f = { }.
Please use the computer for the solution and not write by hand
4. Let X={a,b,c} with the topology r ={X,ø,{a},{b},{c},}{a,b},{a,c},{b,c}}. Verify whether X is connected or not. Explain your answer in details.

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Exercise 1. Let X = {a, b, c, d, e}. Which of the following families is a topology? Why or why not? (1) T= {Ø, {a}, {b}, {a,b}, {d, e}, X}; (2) T= {Ø, {a}, {b}, {a,b}, {d, c}, {a, b, c, d}, X}; (3) T = {0, {a}, {b}, {a,b}, {d, c}, {a,b,c,d}}; (4) T = {0, {a}, {a,b}, {b, c, d, e}, X};