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From the concluding section of Quasi-Polish Spaces by Matthew de Brecht:

"It turns out that the category of quasi-Polish spaces and continuous functions has a very natural description: up to equivalence, it is the smallest full subcategory of the category of topological spaces and continuous functions which contains the Sierpinski space and is closed under countable limits.

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Although quasi-Polish spaces are closed under countable co-products, they are not closed under countable co-limits in general, and the category of quasi-Polish spaces is not cartesian closed."


Am I right in thinking that there is a smallest cartesian closed full subcategory of $\mathrm{Top}$ which contains the Sierpinski space and is closed under both countable limits and countable co-limits? If so, what are some nice characterizations of it? Has it been discussed much?

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My question is answered in Hyperprojective Hierarchy of qcb0-Spaces by Matthias Schröder and Victor Selivanov. The answer is yes (theorem 7.2).

Let $\mathcal { N } = \omega ^ \omega$ be the Baire space. An admissible representation of a space $X$ is a partial continuous surjection $\delta : \subseteq \mathcal { N } \twoheadrightarrow X$ such that every partial continuous map $\nu : \subseteq \mathcal { N } \rightarrow X$ can be continuously reduced to $\delta$, i.e. there exists some continuous $g : \subseteq \mathcal { N } \rightarrow \mathcal { N }$ with $\nu = \delta g$.

For a pointclass $\mathbf { \Gamma }$, the authors denote by $\mathsf { QCB _ 0} ( \mathbf { \Gamma } )$ the collection of spaces with an admissible representation $\delta$ whose defining partial equivalence relation $\{ ( p , q ) \in \mathcal { N } ^ 2 | \delta ( p ) = \delta ( q ) \}$ lies in $\mathbf { \Gamma } ( \mathcal { N } ^ 2 )$. The quasi-Polish spaces are precisely the countably-based spaces in $\mathsf { QCB _ 0 } ( \mathbf { \Pi } ^ 0 _ 2 )$. The category I asked for is $\mathsf { QCB _ 0 } ( \mathbf { HP } ) = \bigcup _ { \alpha < \omega_1 } \mathsf { QCB _ 0 } ( \mathbf { \Sigma } ^ 1 _ \alpha )$; the authors call its objects hyperprojective qcb0-spaces, and it is the main subject of the paper.

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