Q1. Let X = {m, n, p, q}, {ø, X, {p}, {m, n}, {p, q}, {m, n, p}, {q}}and Y = {a, b, c}, F = {ó,Y, {a, b}, {c}}Define f : (X, T) → (Y, F) byf (m) = a, f(n) = b, f(p)Is fa continuous function?Is f an open function?= c, f(q) = c.{ó, X, {c}, {m, n}, {a, b, c, d}, {c, m, n}}.{а, b, c, d, m, п}, т %3D{b, с, т} с X.(a) Is X a connected space? Why?(b) Is X a pathwise connected space? Justify your answer.Q2. Let Xand Y =Q3. Let A = {(x, y), 4x? + 9y² < 36} C R²Is A connected? pathwise connected? Justify your answer.Q4. Let X = {m, n, p, q},T =and Y = {a, b, c, d}, F =Given the mapping f : (X,T) → (Y, F) by{d, X, {p}, {m, n}, {m,n, p}}{ø, X, {d}, {a, b}, {a, b, d}}f(m) == a, f(n) = b, f(p) = d, f(q)= c.Is fa bijection?Is fa continuous mapping?Is f an open mapping?Is fa 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} isnot homeomorphic to the interval [-1, 1].Q6. Given M =(a) Determine dM and ÔM.(b) Is M connected? Pathwise connected? Justify.(c) Let N = {(x, y), x² + y² < 4}.Is N = M ? prove your answer.{(x, y), 1 < x² + y? < 4} c R².1

Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q1. Let X = {m, n, p, q}, {ó, X, {p}, {m, n}, {p, q}, {m, n, p}, {q}} and Y = {a, b, c}, F = {6, Y, {a, b}, {c}} Define f : (X, 7) → (Y, F) by f(m) = a, f(n) = b, f(p) = c, f(q) = c. Is fa continuous function? Is f an open function?

Q1. Let X = {m, n, p, q}, {ó, X, {p}, {m, n}, {p, q}, {m, n, p}, {q}} and Y = {a, b, c}, F = {6, Y, {a, b}, {c}} Define f : (X, 7) → (Y, F) by f(m) = a, f(n) = b, f(p) = c, f(q) = c. Is fa continuous function? Is f an open function?

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

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Q1. Let X = {m, n, p, q}, {ø, X, {p}, {m, n}, {p, q}, {m, n, p}, {q}}
and Y = {a, b, c}, F = {ó,Y, {a, b}, {c}}
Define f : (X, T) → (Y, F) by
f (m) = a, f(n) = b, f(p)
Is fa continuous function?
Is f an open function?
= c, f(q) = c.
{ó, X, {c}, {m, n}, {a, b, c, d}, {c, m, n}}.
{а, b, c, d, m, п}, т %3D
{b, с, т} с X.
(a) Is X a connected space? Why?
(b) Is X a pathwise connected space? Justify your answer.
Q2. Let X
and Y =
Q3. Let A = {(x, y), 4x? + 9y² < 36} C R²
Is A connected? pathwise connected? Justify your answer.
Q4. Let X = {m, n, p, q},T =
and Y = {a, b, c, d}, F =
Given the mapping f : (X,T) → (Y, F) by
{d, X, {p}, {m, n}, {m,n, p}}
{ø, X, {d}, {a, b}, {a, b, d}}
f(m) =
= a, f(n) = b, f(p) = d, f(q)
= c.
Is fa bijection?
Is fa continuous mapping?
Is f an open mapping?
Is fa 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].
Q6. Given M =
(a) Determine dM and ÔM.
(b) Is M connected? Pathwise connected? Justify.
(c) Let N = {(x, y), x² + y² < 4}.
Is N = M ? prove your answer.
{(x, y), 1 < x² + y? < 4} c R².
1

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