any topological space with the fixed point property is connected

Theorem Any topological space with the fixed-point property (http://planetmath.org/FixedPointProperty) is connected.

Proof. We will prove the contrapositive. Suppose $X$ is a topological space which is not connected. So there are non-empty disjoint open sets $A,B\subseteq X$ such that $X=A\cup B$. Then there are elements $a\in A$ and $b\in B$, and we can define a function $f\colon X\to X$ by

 $f(x)=\left\{\begin{array}[]{ll}a,&\mbox{when}\,x\in B,\\ b,&\mbox{when}\,x\in A.\\ \end{array}\right.$

Since $A\cap B=\varnothing$ and $A\cup B=X$, the function $f$ is well-defined. Also, $a\notin B$ and $b\notin A$, so $f$ has no fixed point. Furthermore, if $V$ is an open set in $X$, a short calculation shows that $f^{-1}(V)$ is $\varnothing,A,B$ or $X$, all of which are open sets. So $f$ is continuous, and therefore $X$ does not have the fixed-point property. $\Box$

References

• 1 G.J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.
• 2 L.E. Ward, Topology, An Outline for a First Course, Marcel Dekker, Inc., 1972.
Title any topological space with the fixed point property is connected AnyTopologicalSpaceWithTheFixedPointPropertyIsConnected 2013-03-22 13:56:35 2013-03-22 13:56:35 yark (2760) yark (2760) 12 yark (2760) Theorem msc 47H10 msc 54H25 msc 55M20