Let be a compact metric space, and let be a continuous map satisfying the expansion property:
for all .Prove that T is surjective.
Hey I've difficulty starting this question. How do I go about doing it?
Thanks in advance.
my thought is this:
choose an open cover of ε-balls for X. since X is compact, it has a finite subcover. consider images under T of this subcover. show they form a cover of X.
(hint: for every U in our sub-cover, we have U contained in T(U) because....?)
Another way would be to prove the result by contradiction. Suppose that the point is not in the range of T. The range of T is closed (by compactness) so there exists such that for all x in X.
Now consider the sequence . Use the non-contracting property of T to show that any two points in this sequence are at a distance at least apart. Therefore the sequence cannot have a convergent subsequence, which contradicts the compactness of X.