Problem 16.8 Let X and Y be compact metric spaces and let f: X → Y...

Problem 16.8 Let X and Y be compact metric spaces and let f: X → Y be a continuous onto map with the property that f^-1[{y}] is connected for every y∈Y. Show that ifY is connected then so isX.

Expert Solution

Colby Messinger answered 1 year ago

Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous bijection. Prove...

Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous bijection. Prove that if (X, dX ) is compact, then f is a homeomorphism. (Hint: it might be convenient to use that a function is continuous if and only if the inverse image of every open set is open, if and only if the inverse image of every closed set is closed).

Let X be a compact space and let Y be a Hausdorff space. Let f ∶...

Let X be a compact space and let Y be a Hausdorff space. Let f ∶ X → Y be continuous. Show that the image of any closed set in X under f must also be closed in Y .

Suppose (X, dX) and (Y, dY ) are metric spaces. Define d : (X ×Y )×(X...

Suppose (X, dX) and (Y, dY ) are metric spaces. Define d : (X ×Y )×(X × Y ) → R by d((x, y),(a, b)) = dX(x, a) + dY (y, b). Prove d is a metric on X × Y .

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that...

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset...

Let (X,d) be a metric space. The graph of f : X → R is the...

Let (X,d) be a metric space. The graph of f : X → R is the set {(x, y) E X X Rly = f(x)}. If X is connected and f is continuous, prove that the graph of f is also connected.

18.2.14. Problem. Give examples of metric spaces M and N , a homeomorphism f : M...

18.2.14. Problem. Give examples of metric spaces M and N , a homeomorphism f : M → N , and a Cauchy sequence (xn) in M such that the sequence ?f(xn)? is not Cauchy in N. 18.2.15. Problem. Show that if D is a dense subset of a metric space M and every Cauchy sequence in D converges to a point of M, then M is complete.

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if ...

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if T is One-to-one then T-1 exist on R(T) and T-1 : R(T) à X is also a linear map. 2. Let X, Y and Z be linear spaces over the scalar field F, and let T1 ϵ B (X, Y) and T2 ϵ B (Y, Z). let T1T2(x) = T2(T1x) ∀ x ϵ X. (i) Prove that T1T2 ϵ B (X,Y) is also a...

Let (X, dX) and (Y, dY) be metric spaces.Define the function d : (X × Y...

Let (X, dX) and (Y, dY) be metric spaces.Define the function d : (X × Y ) × (X × Y ) → R by d ((x1, y1), (x2,y2)) = dx(x1,x2)+dy(y1,y2) Prove that d is a metric on X × Y .

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just f (because f is already curried) let f x y z = (x,(y,z)) let f x y z = x (y z)

4. (Oblique Trajectory Problem) Let F(x, y) = x 2 + xy + y 2 ....

4. (Oblique Trajectory Problem) Let F(x, y) = x 2 + xy + y 2 . Find a formula for G(x, y) such that every curve in the one-parameter family defined by F(x, y) = c intersects every curve in the one-parameter family defined by G(x, y) = c at a sixty degree angle

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