Prove that there is no continuous bijection from the unit circle S1 = y21 onto any subset of R. (x,y) E R2 Prove that there is no continuous bijection from the unit circle S1 = y21 onto any subs...
mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism.
mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism.
Find a cone in R2 that is not convex, Prove that a subset X of Rr is a convex cone if and only if x,y eX implies that Xx+ py E X for all
Find a cone in R2 that is not convex, Prove that a subset X of Rr is a convex cone if and only if x,y eX implies that Xx+ py E X for all
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty : 0 ≤ t ≤ 1 } of R n . A subset A of Rn, where n ∈ N, is called convex if it contains the line segment joining any two of its points. It is easy to check that any convex set is path-connected. (a) Let f : X → Y be an...
Let S = {(x, y) = RP.22 + y2 = 1} denote the unit circle in R2 with the subspace topology. Define the function F: (0,1) + S via th (cos(2), sin(24t)) Prove that F is one-to-one, onto, and continuous, but not a homeomorphism.
Let S1 be the unit
circle with the usual topology, S1 × S1 be
the product space, and define the torus T : = [0,1] × [0,1] / ∼ as
a quotient space, where ∼ is the equivalence relation that (1,y) ∼
(0,y) for all y ∈ [0,1] and (x,0) ∼ (x,1) for all x ∈ [0,1]. Prove
that S1 × S1 and T are homeomorphic.
Let Sl be the unit circle with the usual topology, Stx St be the...
{(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2
{(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2
8. a. Prove that f(x) = cos x is continuous on R. b. If ECR and f: E R is continuous on E, prove that 8(x) = cos (f(x)) is continuous on E. 9. For each of the following equations, determine the largest subset E of R such that the given equation
2. Consider the following transformations of R2 Tİ (z, y) (-r, y), T3(x, y) (z, _y), T,(zw) (y, x). Show that, for any j 1,2,3, a subset A C R2 is a Jordan region if and only if T,(A) is a Jordan region. What is the relation between the volumes of A and T, (A)?
2. Consider the following transformations of R2 Tİ (z, y) (-r, y), T3(x, y) (z, _y), T,(zw) (y, x). Show that, for any j 1,2,3,...
1. Consider the unit circle: (x,y) : x2 y2 = 1. T. Let f R2 ->R be defined by f(x,y) = x2-y, and let F : R2 -> R be defined by F(x, y) Compute the integral of f and F around the unit circle. For the integral of F, proceed in the standard (anticlockwise) direction
10. Define φ : R2 → R by φ(x,y-x + y for (x,y) E R2. Show that φ is an onto homomorphism and find the kernel of φ (10 Points)