any topological space with the fixed point property is connected


Theorem Any topological spaceMathworldPlanetmath 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,BX such that X=AB. Then there are elements aA and bB, and we can define a function f:XX by

f(x)={a,whenxB,b,whenxA.

Since AB= and AB=X, the function f is well-defined. Also, aB and bA, so f has no fixed point. Furthermore, if V is an open set in X, a short calculation shows that f-1(V) is ,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, TopologyMathworldPlanetmath and Normed SpacesMathworldPlanetmath, 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