The question we received is as follows:
Show that an open set in $\mathbb{R}^n$ cannot have the fixed point property
where A has the "fixed point property" if every continuous function from A to itself has a fixed point (for example $D^n$ has the fixed point property - Brouwer's Theorem)
I have seen that someone has already asked this for an open ball - where that is solved due to homeorphism to $\mathbb{R}^n$. Can this be generalized to every open set?
My attempt was to find a function that slightly moves each point - each points has a ball around it that is still in the set, but I'm having trouble showing this can be done continuously.