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.
{(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
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
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...
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...
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,...
2. Įpp. 492, Marsden & Hoffman Let y : [a,b] → R and ψ : R → R be continuous. Show that A = {(x,o(x)) : x [a,아 C R2 has volume zero in R2 and the set B-{(x, ψ (x)) : x E R} C R2 has measure zero in IK. 2. Įpp. 492, Marsden & Hoffman Let y : [a,b] → R and ψ : R → R be continuous. Show that A = {(x,o(x)) : x [a,아...
(2) (a) Prove that there is a C1 map u : E → R-defined in a neighborhood E c R2 of the point (1,0) such that (b) Find u'(x) for x E E (c) Prove that there is a Cl map : G → R2 defined in a neighborhood G C R2 of the point (1,0) such that for all y EG (2) (a) Prove that there is a C1 map u : E → R-defined in a neighborhood E...
Prove d2 is a metric d2 : R2 x R2 - R (r, y) max{a 2 - y2}.