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There is an interesting correspondence between notions of topology and notions of computability theory originating from the ingenious idea of Dana Scott to identify computable functions with continuous functions (in fact it can perhaps be traced back to Brouwer).

Complexity theory can be seen as a refinement of computability theory: we are interested in not only whether a problem is solvable or not but also the efficiency with which it is solvable. Taking this view of complexity theory as a refinement of computability theory, I wonder if there is any research on analogous mathematical structures to provide efficiency-sensitive denotational semantics to programming languages.

The point of topology is that it is distance-blind which is in accord with the efficiency-blind approach in computability theory. So it seems to me that an efficiency-sensitive denotational structure should be a topological space with a notion of distance, i.e., a metric space or something like that.

Has someone worked this out or is this a nonsensical question? If it is the latter, please explain why.

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