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Q2. Let X = {a, b, c, d, m, n}, T = {ø, X, {c}, {m, n}, {a, b, c, d}, {c, m, n}}. and Y = {b, c, m} C X. (a) Is X a connected space? Why? (b) Is X a pathwise connected space? Justify your answer. 26) c D2

Q2. Let X = {a, b, c, d, m, n}, T = {ø, X, {c}, {m, n}, {a, b, c, d}, {c, m, n}}. and Y = {b, c, m} C X. (a) Is X a connected space? Why? (b) Is X a pathwise connected space? Justify your answer. 26) c D2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please answer Q2, Q3, Q4, Q5 and Q6 plz
Q2. Let X = {a, b, c, d, m, n}, T = {ø, X, {c}, {m, n}, {a, b, c, d}, {c, m, n}}. and Y = {b, c, m} C X. (a) Is X a connected space? Why? (b) Is X a pathwise connected space? Justify your answer. Q3. Let A = {(x, y), 4x² + 9y² < 36} C R² Is A connected? pathwise connected? Justify your answer. Q4. Let X = {m, n, p, q}, T = {ø, X, {p}, {m,n}, {m, n, p}} and Y = {a, b, c, d}, F = {ø, X, {d}, {a, b}, {a, b, d}} Given the mapping f: (X,7)· (Y, F) by f(m) = a, f(n) = b, f(p) = d, f(q) = c. %3D Is fa bijection? Is fa continuous mapping? Is f an open mapping? Is f a homeomorphism ? What you can say about the topological spaces (X,T) and (Y, F). ( Show the details of your answers). Q5. Prove that the circle {(x, y), x² + y² = 1} is not homeomorphic to the interval [-1, 1]. %3D Q6. Given M = {(x, y), 1 < x² + y² < 4} C R². (a) Determine dM and aM. (b) Is M connected? Pathwise connected? Justify. (c) Let N = {(x, y), x² + y² < 4}. Is N = M ? prove your answer.
Transcribed Image Text:Q2. Let X = {a, b, c, d, m, n}, T = {ø, X, {c}, {m, n}, {a, b, c, d}, {c, m, n}}. and Y = {b, c, m} C X. (a) Is X a connected space? Why? (b) Is X a pathwise connected space? Justify your answer. Q3. Let A = {(x, y), 4x² + 9y² < 36} C R² Is A connected? pathwise connected? Justify your answer. Q4. Let X = {m, n, p, q}, T = {ø, X, {p}, {m,n}, {m, n, p}} and Y = {a, b, c, d}, F = {ø, X, {d}, {a, b}, {a, b, d}} Given the mapping f: (X,7)· (Y, F) by f(m) = a, f(n) = b, f(p) = d, f(q) = c. %3D Is fa bijection? Is fa continuous mapping? Is f an open mapping? Is f a homeomorphism ? What you can say about the topological spaces (X,T) and (Y, F). ( Show the details of your answers). Q5. Prove that the circle {(x, y), x² + y² = 1} is not homeomorphic to the interval [-1, 1]. %3D Q6. Given M = {(x, y), 1 < x² + y² < 4} C R². (a) Determine dM and aM. (b) Is M connected? Pathwise connected? Justify. (c) Let N = {(x, y), x² + y² < 4}. Is N = M ? prove your answer.
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