
3. (a) Let (X, dx), (Y, dy) be two metric spaces, C C X connected, and...
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.
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
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
For metric spaces (X, dx) and (Y, dy) consider their Cartesian product Z-X (p18). Show that the following constructions both give metris on the product (a) Define di : Z × c, d))-dr(a, c) + dy(b, d) for (a, b), (c, d) e X x Y (b) Define (lo : Z × Z → R by writing do ((a, b), (c, d))-maux {dx (a, c), dy(b, d)) for (a, b), (c, d) E X × Y Answer the following: (c)...
9. Let X and Y be metric spaces, and let D be a dense subset of X. (For the definition of "dense, see Problem 4 at the end of Section 3.5.) (a) Let f : X → Y and g : X → Y be continuous functions. Suppose that f(d)gld) for all d E D. Prove that f and g are the same function.
(TOPOLOGY) Prove the following using the defintion:
Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...
Let R^2 be equipped by the metric ds^2 = (4/(1 + x^2 + y^2 )^2)
(dx^2 + dy^2 ), i.e. its first fundamental form is E = G =
4/(1+x^2+y^2)^2 , F = 0. Use the formula dω12 = −Kω1 ∧ ω2 to
calculate its Gauss curvature.
Let R2 be equipped by the metric i.e. its first fundamental form is E = G = TrtFF, F = 0. Use the formula dui,-- . КМ Л w2 to calculate its Gauss...
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)