TITLE: Does a topological space always admit a $\mathbb{Z}_2$-action?
QUESTION [6 upvotes]: Recently I have been reading a lot about $\mathbb{Z}_2$-actions on topological spaces. Mainly I was focused on surfaces such as the sphere, torus and Klein bottle and here the existence of a nontrivial $\mathbb{Z}_2$-action is rather simple. But I was wondering if a general topological space always admits a nontrivial continuous $\mathbb{Z}_2$-action? If not, then more specific, does a manifold always admit a nontrivial continuous $\mathbb{Z}_2$-action? 
For a manifold $M$ I was thinking about the fact that we can embed $M$ into $\mathbb{R}^N$ for some $N >0$ and then $M$ can inherit a $\mathbb{Z}_2$-action from $\mathbb{R}^N$ but then when one looks at the spiral in $\mathbb{R}^2$ we see that this spiral does not inherit for example the antipodality of $\mathbb{R}^2$.
Extra: I was also wondering that if there are spaces that do admit a nontrivial continuous $\mathbb{Z}_2$-action, do these space then also admit a free $\mathbb{Z}_2$-action? By free I mean that the action is fixed point free.
If anyone knows some basic examples that do not admit a continuous (free) $\mathbb{Z}_2$-action. Please do share. I seem to be unable to find one.
Thank you in advance!

REPLY [4 votes]: $X=\mathbb R$ easily admits a nontrivial $\mathbb Z_2$-action, but it cannot be free -- we can always find a fixed point with the intermediate value theorem.