Analogue of the topology-computability correspondence for computational complexity 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.
The answer to your literal question is neither. The reasoning in your question makes perfect sense, and there are several researchers working on the details on what exactly is analogous to complexity theory in the same way topology is analagous to computability. However, I don't think I am unfair in claiming that this is nowhere near completed.
In a compact metric context, notions such as metric entropy seem to be important, and the counterpart to &quot;continuous&quot; seems to be &quot;having a modulus of continuity of particular growth&quot;. Beyond compact metric, the picture quickly gets very unclear.
I don't know of a comprehensive written treatment, but Martin Ziegler just gave a tutorial on this at the conference &quot;Computability in Europe&quot;. The talks are up on this Youtube channel, specifically &quot;Quantitative Coding and Complexity theory...&quot;.