A space $(X,\tau)$ is called *rigid* if $\textrm{Aut}(X)=\{\textrm{id}\_X\}$. We say $(X,\tau)$ is *strongly rigid* if for every continuous map $f:X\to X$ we have that $f = \textrm{id}\_X$ or $f$ is constant (that is there is $x\_0\in X$ such that $f(x)=x\_0$ for all $x\in X$).


Is there a strongly rigid Hausdorff space with more than 1 element?



Strongly rigid Hausdorff spaces
Yes, see for example "[Continua which admit only the identity mapping onto non-degenerate subcontinua](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv60i1p23bwm)" by H. Cook (Fund. Math. 60, 1967, 241-249).


