# A continuum-sized convenient category of topological spaces

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?

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.