any topological space with the fixed point property is connected

TheoremAny 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 $A,B\subseteq X$ suchthat $X=A\cup B$. Then there are elements $a\in A$ and $b\in B$, andwe can define a function $f:X\to X$ by

Since $A\cap B=\mathrm{\varnothing }$ 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 $\mathrm{\varnothing },A,B$ or $X$, all of which are open sets.So $f$ is continuous, and therefore $X$ does not have the fixed-point property.$\mathrm{\square }$