fixed point property
The fixed point property is obviously preserved under homeomorphisms. If is a homeomorphism between topological spaces and , and has the fixed point property, and is continuous, then has a fixed point , and is a fixed point of .
A space with only one point has the fixed point property.
A closed interval of has the fixed point property. This can be seen using the mean value theorem. (http://planetmath.org/BrouwerFixedPointInOneDimension)
The extended real numbers have the fixed point property, as they are homeomorphic to .
The topologist’s sine curve has the fixed point property.
The real numbers do not have the fixed point property. For example, the map on has no fixed point.
An open interval of does not have the fixed point property. This follows since any such interval is homeomorphic to . Similarly, an open ball in does not have the fixed point property.
For each , the real projective space has the fixed point property.
Every simply-connected plane continuum has the fixed-point property.
The Alexandroff–Urysohn square (also known as the Alexandroff square) has the fixed point property.
Any topological space with the fixed point property is connected (http://planetmath.org/AnyTopologicalSpaceWithTheFixedPointPropertyIsConnected) and (http://planetmath.org/T0Space).
Suppose is a topological space with the fixed point property, and is a retract of . Then has the fixed point property.
Suppose and are topological spaces, and has the fixed point property. Then and have the fixed point property. (Proof: If is continuous, then is continuous, so has a fixed point.)
- 1 G. L. Naber, Topological methods in Euclidean spaces, Cambridge University Press, 1980.
- 2 G. J. Jameson, Topology and Normed Spaces, 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|
|Date of creation||2013-03-22 13:56:32|
|Last modified on||2013-03-22 13:56:32|
|Last modified by||yark (2760)|