Analogue of the topology-computability correspondence for computational complexity
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;.