Are there any path-connected sets (of $\Bbb R^2$) that guarantee two or more fixed points for any continuous bijections mapping them onto themselves?

Question

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!

Answer 1

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