fixed point property

fixed point property

Let

X

be a

topological space

.
If every

continuous function

f

:

X

→

X

has a

fixed point

(

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

:

X

→

Y

is a homeomorphism between topological
spaces

X

and

Y

, and

X

has the fixed point property,
and

f

:

Y

→

Y

is

continuous

,
then

h

-

1

∘

f

∘

h

has a fixed point

x

∈

X

,
and

h

⁢

(

x

)

is a fixed point of

f

.

Examples

1.

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

2.

A

closed interval

[

a

,

b

]

of

ℝ

has the fixed point property.
This can be seen using the mean value theorem. (

http://planetmath.org/BrouwerFixedPointInOneDimension

)

3.

The extended real numbers have the fixed point property,
as they are homeomorphic to

[

0

,

1

]

.

4.

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

5.

The

real numbers

ℝ

do not have the fixed point property.
For example, the map

x

↦

x

+

1

on

ℝ

has no fixed point.

6.

An open interval

(

a

,

b

)

of

ℝ

does not have the fixed point property.
This follows since any such interval is homeomorphic to

ℝ

.
Similarly, an

open ball

in

ℝ

n

does not have the fixed point property.

7.

Brouwer’s

Fixed Point Theorem

states that in

ℝ

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 Theorem

generalizes this result further.

8.

For each

n

∈

ℕ

, the real projective space

ℝ

⁢

ℙ

2

⁢

n

has the fixed point property.

9.

Every simply-connected plane

continuum

has the fixed-point property.

10.

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

Properties

1.

Any topological space with the fixed point property is

connected

(

http://planetmath.org/AnyTopologicalSpaceWithTheFixedPointPropertyIsConnected

)
and

T

0

(

http://planetmath.org/T0Space

).

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.

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

:

X

→

X

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,

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

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

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