Is the singleton space the only Hausdorff space $X$ such that the set of automorphisms $\varphi: X\to X$ equals $\{\textrm{id}\_X\}$?



Hausdorff spaces with trivial automorphism group
Not at all. Those spaces are called *rigid* and there are plenty of examples in the literature. The opposite notion is *homogeneity* which is a better studied property. The first (non-trivial) rigid space was constructed by Kuratowski in "[Sur la puissance de l'ensemble des nombres de dimension au sens de M Frechet](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv8i1p15bwm)" (Fund.Math 8, 1926, 201-208). I also recommend "[Decompositions of rigid spaces](http://www.ams.org/journals/proc/1983-089-03/S0002-9939-1983-0715881-1/)" by van Engelen and van Mill (PAMS 89, 1983, 533-536), where they give two nice examples: a rigid space that can be decomposed into two homeomorphic homogeneous subspaces and a homogeneous space that can be decomposed into two homeomorphic rigid subspaces. By the way, these two examples are even separable and metrizable.


