Editing Fixed point (mathematics) (section)

==Topological fixed point property==
{{main article|Fixed-point property}}

A [[topological space]] <math>X</math> is said to have the [[fixed point property]] (FPP) if for any [[continuous function]]
:<math>f\colon X \to X</math>
there exists <math>x \in X</math> such that <math>f(x)=x</math>.

The FPP is a [[topological invariant]], i.e., it is preserved by any [[homeomorphism]]. The FPP is also preserved by any [[Retraction (topology)|retraction]].

According to the [[Brouwer fixed-point theorem]], every [[compact space|compact]] and [[convex set|convex]] [[subset]] of a [[Euclidean space]] has the FPP. Compactness alone does not imply the FPP, and convexity is not even a topological property, so it makes sense to ask how to topologically characterize the FPP. In 1932 [[Karol Borsuk|Borsuk]] asked whether compactness together with [[contractible space|contractibility]] could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita, who found an example of a compact contractible space without the FPP.<ref>{{cite journal |last=Kinoshita |first=Shin'ichi |year=1953 |title=On Some Contractible Continua without Fixed Point Property |journal=[[Fundamenta Mathematicae|Fund. Math.]] |volume=40 |issue=1 |pages=96–98 |doi=10.4064/fm-40-1-96-98 |issn=0016-2736 |doi-access=free}}</ref>