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We know by Brouwer‘s fixed-point theorem that any continuous bijection mapping the closed unit circle to itself must have a fixed point.

My question: are there any path-connected sets (preferably subsets of $\mathbb R^2$) that guarantee two or more fixed points for any continuous bijections mapping them onto themselves?

The reason I‘m imposing the path-connectedness restriction is that it‘s easy to come up with a “trivial” example by taking the union of two non-homeomorphic sets with the fixed-point property (such as the union of $[0,1]$ with the closed unit circle).

Thanks!

Hanul Jeon
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Franklin Pezzuti Dyer
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1 Answers1

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Try the union of the unit circle and the interval $[1,2]$ on the $x$ axis. Any homeomorphism of this must fix $1$ and $2$.

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Robert Israel
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  • Beat me to it by just a few seconds! – John Hughes Jun 24 '20 at 16:11
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    @RobertIsrael Ah, cool. I suppose if we wanted to make a space with $n$ fixed points, we could make something star-shaped and “glue” a different object to each point of the star, so that none of them can be mapped on to each other. – Franklin Pezzuti Dyer Jun 24 '20 at 16:12