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
Canonical name	AnyTopologicalSpaceWithTheFixedPointPropertyIsConnected
Date of creation	2013-03-22 13:56:35
Last modified on	2013-03-22 13:56:35
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	12
Author	yark (2760)
Entry type	Theorem
Classification	msc 47H10
Classification	msc 54H25
Classification	msc 55M20

any topological space with the fixed point property is connected

References

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