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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?

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Yes, see for example "Continua which admit only the identity mapping onto non-degenerate subcontinua" by H. Cook (Fund. Math. 60, 1967, 241-249).

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