@mcc almost yes.
it's really just missing a way to compute partial things in parallel (i.e. dovetailing). then it would be perfect, e.g. with the (partial) unit type representing Sierpinski space S, the "open sets" of a type T are in correspondence with partial functions T -> S, and you get the union of open sets by dovetailing.
but you can do a lot of interesting stuff without dovetailing. e.g. see "seemingly impossible functional programs": https://math.andrej.com/2007/09/28/seemingly-impossible-functional-programs/
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