Any help with these would be hugely appreciated
(1) Let (X, d) and (Y, d'
) be metric spaces, and let f : X — » Y be continuous with f(X) = Y. Show that if (X,d) is complete and
d(x,y) < kd'(f(x),f(y)) for some constant k and all x,y € X, then (Y, d') is complete.
(2) Let A be a non-empty compact subset of X. Prove that there exist points a,b € A such that d(a, b) = sup{d(x,y) : x,y € A}.
(3)(a) Prove that a topological space X is connected if and only if every continuous map f : X —» {0,1} is constant, where {0,1} is equipped with the discrete topology.
(b) Let S
1 be the unit circle {(x,y) € R2 : x2 + y2 = 1} in R2 with the topology induced by the standard topology of R2 . Assume that
f : S1 —» R is a continuous map. Prove that there exists a point
z = (x,y) € S1 such that f(z) = f(-z). (Hint: Consider the function
g(z) = f(z) - f(-z)).