Topological Fixed Point Property
A topological space is said to have the fixed point property (briefly FPP) if for any continuous function
there exists such that .
The FPP is a topological invariant, i.e. 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.
Read more about this topic: Fixed Point (mathematics)
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