I'm trying to find if the following has the fixed point property:
Is it enough to show:
Let
so belongs to but
Thus, does not have the fixed point property.
then this would depend on the structure imposed on S. for example, if one is regarding S as a topological group, then sure, because then (1,0) has to be a fixed point. but if one is just regarding S as a topological space, then no, we can pick any rotation by an angle that is not an integral multiple of