Theorem Any topological space with the fixed-point property 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 $A\cap B=\emptyset$ 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 $\emptyset, A,B$ or $X$ , all of which are open sets. So $f$ is continuous, and therefore $X$ does not have the fixed-point property.

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.