I'm trying to find if the following has the fixed point property:

$\displaystyle S= \{ (x,y) \, : \, x^2+y^2=1 \}$

Is it enough to show:

Let $\displaystyle f \, : S \rightarrow S$

$\displaystyle 0^2 + (-1)^2=1 \,$ so $\displaystyle (0,-1)$ belongs to $\displaystyle S$ but $\displaystyle f(0,-1) = (0,1)$

Thus, $\displaystyle S$ does not have the fixed point property.