If $(X,\tau)$ is a topological space, we denote by $\text{Cont}(X,X)$ the collection of all continous functions $f:X\to X$. We say that $(X,\tau)$ has the *fixed point property* if for any $f\in\text{Cont}(X,X)$ there is $x\_0\in X$ such that $f(x\_0) = x\_0$.


If $(X,\tau)$ has the fixed point property, does $\text{Cont}(X,X)$ have the fixed point property?



If $X$ has the fixed point property, what about $\text{Cont}(X,X)$?
No. In infinite dimensional normed spaces, [there exists a retraction of the unit Ball onto the unit sphere](http://www.ams.org/journals/proc/1983-088-03/S0002-9939-1983-0699410-7/S0002-9939-1983-0699410-7.pdf). Since for $X=[-1,1]$ with the usual topology, $C(X,X)$ is simply the unit Ball of $C(X)$ and since the unit sphere of the latter admits a trivial fixed point free self map (multiplication with $-1$), we are done. 


