For metric spaces (X, dx) and (Y, dy) consider their Cartesian product Z-X (p18). Show that...
2. Let (X, dx), (Y, dy) be two metric spaces, and f:X + Y a map. (a) Define what it means for the map f to be continuous at a point x E X. (b) Suppose W X is compact. Prove that then f(W) CY is compact.
3. (a) Let (X, dx), (Y, dy) be two metric spaces, C C X connected, and f : X+Y continuous. Show that f(C) CY is connected.
2. Let (X, dx), (Y, dy), (2, dz) be metric spaces, and f : XY,g:Y + Z continu- ous maps. (a) Prove that the composition go f is continuous. (b) Prove that if W X is connected, then f(W) CY is connected.
Let (X, dx), (Y, dy) be metric spaces and fn be a sequence of functions fn: XY Prove that if {fn} converges uniformly on X then for any a є x lim lim fn()- lim lim /) xa n-00 Let (X, dx), (Y, dy) be metric spaces and fn be a sequence of functions fn: XY Prove that if {fn} converges uniformly on X then for any a є x lim lim fn()- lim lim /) xa n-00
show that the product metric space X and Y are topologically equivalent 2. Suppose that metric space (X, d) is topologically equivalent to (Y, dh) for i-, n. Show that the product metric spaces X = Π-x, and Y = Π, Y, are topologica equivalent. 2. Suppose that metric space (X, d) is topologically equivalent to (Y, dh) for i-, n. Show that the product metric spaces X = Π-x, and Y = Π, Y, are topologica equivalent.
x dx dy + y) dx dy 0 (b (d a)(c) Answer: (a) x dx dy + y) dx dy 0 (b (d a)(c) Answer: (a)
Fx 0. Show that =-- dx Fy dy 8. Suppose y is a function of z, F(x, y) = 0, and F,メO. Show that dr--Fr 9. Fid the critical points of f(z, y) if any exist, for (a, y) = ex sin y 10. Calculate the iterated integral: ysin(zy)d dy Fx 0. Show that =-- dx Fy dy 8. Suppose y is a function of z, F(x, y) = 0, and F,メO. Show that dr--Fr 9. Fid the critical points...
For each of the following metric spaces (X, d) and subsets A S X decide whether A is open, closed, neither or both. You do not need to justify your answers. (a) X-Z, d is the discrete metric. ACX is any subset (b) X = R2, d = d2 is the Euclidean metric. A = {(x, yje R2 : x-y) (c) X = R2, d = d2 is the Euclidean metric. A (0, 1) × {0).
For metric spaces and topology Problem II. a) Show that f: X →Y is continuous if and only if f-'(C) CX is closed for every closed C CY b) Then show that a function f: X + Y is continuous if and only if f(A) < f(A) for all ACX
Foundations of Analysis Let X = {A, B,C,D}. ·Describe a function δ : X × X-> R that is a metric on X. . Describe a function δ : X x X-R that satisfies δ(z,x)-0 and δ(x,y)-δ(y,x) for all z, y E X but that is NOT a metric Let X = {A, B,C,D}. ·Describe a function δ : X × X-> R that is a metric on X. . Describe a function δ : X x X-R that satisfies...