any topological space with the fixed point property is connected
Proof. We will prove the contrapositive. Suppose is a topological space which is not connected. So there are non-empty disjoint open sets such that . Then there are elements and , and we can define a function by
Since and , the function is well-defined. Also, and , so has no fixed point. Furthermore, if is an open set in , a short calculation shows that is or , all of which are open sets. So is continuous, and therefore does not have the fixed-point property.
|Title||any topological space with the fixed point property is connected|
|Date of creation||2013-03-22 13:56:35|
|Last modified on||2013-03-22 13:56:35|
|Last modified by||yark (2760)|