fixed point property


Let X be a topological spaceMathworldPlanetmath. If every continuous functionMathworldPlanetmath f:XX has a fixed pointMathworldPlanetmath (http://planetmath.org/FixedPoint), then X is said to have the fixed point property.

The fixed point property is obviously preserved under homeomorphisms. If h:XY is a homeomorphism between topological spaces X and Y, and X has the fixed point property, and f:YY is continuous, then h-1fh has a fixed point xX, and h(x) is a fixed point of f.

Examples

  1. 1.

    A space with only one point has the fixed point property.

  2. 2.

    A closed intervalMathworldPlanetmath [a,b] of has the fixed point property. This can be seen using the mean value theorem. (http://planetmath.org/BrouwerFixedPointInOneDimension)

  3. 3.

    The extended real numbers have the fixed point property, as they are homeomorphic to [0,1].

  4. 4.

    The topologist’s sine curve has the fixed point property.

  5. 5.

    The real numbers do not have the fixed point property. For example, the map xx+1 on has no fixed point.

  6. 6.

    An open interval (a,b) of does not have the fixed point property. This follows since any such interval is homeomorphic to . Similarly, an open ballPlanetmathPlanetmath in n does not have the fixed point property.

  7. 7.

    Brouwer’s Fixed Point Theorem states that in n, the closed unit ball with the subspace topology has the fixed point property. (Equivalently, [0,1]n has the fixed point property.) The Schauder Fixed Point TheoremMathworldPlanetmath generalizes this result further.

  8. 8.

    For each n, the real projective space 2n has the fixed point property.

  9. 9.

    Every simply-connected plane continuum has the fixed-point property.

  10. 10.

    The Alexandroff–Urysohn square (also known as the Alexandroff square) has the fixed point property.

Properties

  1. 1.

    Any topological space with the fixed point property is connectedPlanetmathPlanetmath (http://planetmath.org/AnyTopologicalSpaceWithTheFixedPointPropertyIsConnected) and T0 (http://planetmath.org/T0Space).

  2. 2.

    Suppose X is a topological space with the fixed point property, and Y is a retract of X. Then Y has the fixed point property.

  3. 3.

    Suppose X and Y are topological spaces, and X×Y has the fixed point property. Then X and Y have the fixed point property. (Proof: If f:XX is continuous, then (x,y)(f(x),y) is continuous, so f has a fixed point.)

References

  • 1 G. L. Naber, Topological methods in Euclidean spaces, Cambridge University Press, 1980.
  • 2 G. J. Jameson, TopologyMathworldPlanetmath and Normed SpacesMathworldPlanetmath, Chapman and Hall, 1974.
  • 3 L. E. Ward, Topology, An Outline for a First Course, Marcel Dekker, Inc., 1972.
  • 4 Charles Hagopian, The Fixed-Point Property for simply-connected plane continua, Trans. Amer. Math. Soc. 348 (1996) 4525–4548.
Title fixed point property
Canonical name FixedPointProperty
Date of creation 2013-03-22 13:56:32
Last modified on 2013-03-22 13:56:32
Owner yark (2760)
Last modified by yark (2760)
Numerical id 20
Author yark (2760)
Entry type Definition
Classification msc 55M20
Classification msc 54H25
Classification msc 47H10
Synonym fixed-point property
Related topic FixedPoint