Fixed point (mathematics)
Topological fixed point property

A topological space X {\displaystyle X} is said to have the fixed point property (FPP) if for any continuous function

there exists x ∈ X {\displaystyle x\in X} such that f (x) = x {\displaystyle f(x)=x}.

The FPP is a topological invariant, i.e., it is preserved by any homeomorphism. The FPP is also preserved by any retraction.

According to the Brouwer fixed-point theorem, every compact and 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 Borsuk asked whether compactness together with 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.
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