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If $X, Y$ are topological spaces, let $\newcommand{\Cont}{\text{Cont}}\Cont(X,Y)$ denote the collection of continous maps $f:X\to Y$, and we endow $\Cont(X,Y)$ with the product topology inherited from $Y^X$.

Question. Is there a topological space $(X,\tau)$ with $\tau\neq\{\emptyset, X\}$ such that there is a topological space $Z$ with $\Cont(X,Z) \cong X$?

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    $\begingroup$ I dont think so. What if X is the two point space with the discrete topology. Then for any Z the continous maps are in bijection with ordinary maps. Cont(X,Z) has either one element (if Z has one element), or a cardinality strictly larger than 2 if Z has more than one element. $\endgroup$
    – Thomas Rot
    Commented May 8 at 13:11
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    $\begingroup$ I think the question has its quantifiers change, but after the edit the answer should be yes. Using domain theory, one can produce a continuous domain $D$ such that $D \cong (D \to D)$ and the category of continuous domains embeds as a full subcategory into the category of topological spaces $\endgroup$
    – daniel gratzer
    Commented May 8 at 20:02

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