injective map between real numbers is a homeomorphism
Proof. Since is injective, then of course is monotonic. Without loss of generality, we may assume that is increasing. Let . Since is open, then there are such that . Therefore and (because continuous functions are Darboux functions) for any there exists such that . This shows that is an open neighbourhood of contained in and therefore (since was arbitrary) is open.
Proof. Of course, it is enough to show that is an open map. But if is open, then there are disjoint, open intervals such that
is open. This shows that is a homeomorphism onto image.
|Title||injective map between real numbers is a homeomorphism|
|Date of creation||2013-03-22 18:53:58|
|Last modified on||2013-03-22 18:53:58|
|Last modified by||joking (16130)|