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!
from Hot Weekly Questions - Mathematics Stack Exchange
Franklin Pezzuti Dyer
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