(a) Let X1 and X2 be metric spaces and suppose f : X1 → X2 is bijective. If X1 is compact and f is continuous on X1, prove that f–1 : X2 → X1 is continuous on X2.(b) Show that the compactness of X1 is necessary in part (a) by finding a continuous bijection f from [0, 2π) onto the unit circle C in R2 such that f–1 is not continuous on C.

(a) Let X 1 and X 2 be metric spaces and suppose f : X 1 → X 2 is bijective. If X 1 is compact and f is continuous on X 1 , prove that f –1 : X 2 → X 1 is continuous on X 2 . (b) Show that the...

(a) Let X₁
and X₂
be metric spaces and suppose f : X₁
→ X₂
is bijective. If X₁
is compact and f is continuous on X₁, prove that f^–1
: X₂
→ X₁
is continuous on X₂.

(b) Show that the compactness of X₁
is necessary in part (a) by finding a continuous bijection f from [0, 2π) onto the unit circle C in R₂
such that f^–1
is not continuous on C.

May 05, 2022

SOLUTION.PDF

(a) Let X 1 and X 2 be metric spaces and suppose f : X 1 → X 2 is bijective. If X 1 is compact and f is continuous on X 1 , prove that f –1 : X 2 → X 1 is continuous on X 2 . (b) Show that the...

Get Answer To This Question

Related Questions & Answers

Submit New Assignment