We say that a space $(X,\tau)$ has the fixed point property (FPP) if for every continuous map $f:X\to X$ there is $x\in X$ with $f(x) = x$.
What is an example of a space $X$ with FPP such that $X^2$ (with the product topology) does not have FPP?
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asked Oct 20, 2017 at 7:49
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$\begingroup$ jstor.org/stable/2033633?seq=1#page_scan_tab_contents example 2 here $\endgroup$– Fedor PetrovCommented Oct 20, 2017 at 8:14
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2 Answers
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The topological space $$ X = \{ (x,\sin \left( \frac{\pi}{1-x} \right)) \ | \ 0 \leq x < 1 \} \cup {(1,1)} \subseteq \mathbb{R}^2 $$ has the FPP but its square does not. See Example $2$ p.$977$ of E. H. Connell, Properties of Fixed Point Spaces.
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$\begingroup$ Fedor Petrov was faster than myself :) $\endgroup$– js21Commented Oct 20, 2017 at 8:17
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The examples given by Petrov and js21 work for the category of metric spaces.
W. Lopez found in "An example in the fixed point theory of polyhedra. Bull. Amer. Math. Soc. 73(1967), 922-924" a compact polyhedron $K$ with the FPP whose square $K^2$ lacks this property.
I. Sadofschi Costa proves in "Presentation complexes with the fixed point property. Geometry and Topology 21(2017), 1275–1283" that $K$ above can be taken of dimension 2.
If $X$ is a finite topological space with the FPP, $X^2$ has the FPP. This was proved by Roddy in "Fixed points and products, Order 11(1994), 11–14".
For $A$-spaces (arbitrary intersections of open sets is open), I believe this is an open question.
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$\begingroup$ What do you call the "category of metric spaces"? I have many natural examples of arrows (continuous, Lipschitz, 1-Lipschitz, etc) $\endgroup$– YCorCommented Mar 26, 2018 at 22:26
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$\begingroup$ The morphisms are the continuous maps, in all the examples. $\endgroup$ Commented Mar 27, 2018 at 16:30
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$\begingroup$ "metrizable space" is then a good word $\endgroup$– YCorCommented Mar 27, 2018 at 18:04