Is there an infinite $T\_2$-space $X$ with $X\cong \text{Aut}(X)$? (Here, $\text{Aut}(X)$ is the set of automorphisms $\varphi:X\to X$ and it carries the topology inherited from the product topology on $X^X$.)


What's the answer if we endow $\text{Aut}(X)$ with the compact-open topology instead of the product topology?



$T_2$-space $X$ with $X\cong \text{Aut}(X)$
In [this paper](http://www.ams.org/journals/tran/1983-280-02/S0002-9947-1983-0716833-2/S0002-9947-1983-0716833-2.pdf) Jan van Mill has constructed a separable metrizable Boolean topological group $X$ such that each homeomorphism $h:X\to X$ 
is equal to the translation $X\to X$, $x\mapsto x+h(0)$. This implies that the homomorphism group $Aut(X)$ of $X$ (endowed with any topology intermediate between the topology of pointwise convergence and the topology of uniform convergence) is topologically isomorphic (and hence homeomorphic) to $X$.


