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Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following properties:

i) for any topological space $S\in\operatorname{Top}$, there exists another topological space $T\in\mathcal{U}$ such that $S$ is isomorphic to some subspace of $T$;

ii) for any topological space $Q\in\operatorname{Top}$, there exists another topological space $R\in\mathcal{V}$ such that $Q$ is isomorphic to some quotient of $R$.

We know that the classes $\operatorname{Comp}$ of compact spaces and $\operatorname{Haus}$ of Hausdorff spaces satisfy the conditions i) and ii), respectively. I want to know if these classes are minimal with respect to this property, that is, if $\operatorname{Comp}\subset\mathcal{U}$ (resp. $\operatorname{Haus}\subset\mathcal{V}$), for all $\mathcal{U}$ (resp. for all $\mathcal{V}$).

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    $\begingroup$ Take $\mathcal{U} = \mathcal{V} =$ the class of all infinite topological spaces. Counterexample to both i) and ii). $\endgroup$
    – Nik Weaver
    Aug 28, 2019 at 16:33

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