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).

Expert Solution

Colby Messinger answered 2 years ago

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 .

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 .

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.

Topology question: Prove that a bijection f : X → Y is a homeomorphism if and...

Topology question: Prove that a bijection f : X → Y is a homeomorphism if and only if f and f−1 map closed sets to closed sets.

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...

dy/dx + y/x \ x3y2

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is...

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is continuous on [a, b]

dy/dx+(y/2x)=(x/(y^3))

Let X, Y be two topological spaces. Prove that if both are T1 or T2 then...

Let X, Y be two topological spaces. Prove that if both are T1 or T2 then X × Y is the same in the product topology. Prove or find a counterexample for T0.

6.3.8. Problem. Let f : A → B be a continuous bijection between subsets of R....

6.3.8. Problem. Let f : A → B be a continuous bijection between subsets of R. (a) Show by example that f need not be a homeomorphism. (b) Show that if A is compact, then f must be a homeomorphism. 6.3.9. Problem. Find in Q a set which is both relatively closed and bounded but which is not compact.

Question