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Brouwer’s fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least ones fixed point. If we replace the convex condition with let's say connectedness (i.e. there is no separation by two disjoint open sets) then what space and map would have no fixed points? I've trying to think of subsets of $\mathbb{R}^2$ that I can do this with, but nothing comes to mind. Any ideas?

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The classical example: the unit circle, and rotation by some nonzero angle.

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