I mean the notion of compactness is inspired pretty clearly from the continuum. I also have a degree in pure math and disagree strongly with your characterization of topology, which is literally defined up to homeomorphism which is itself defined by it's continuity. Arguably the whole point of topology is characterized by that which is preserved by continuous deformation, which clearly is inspired by more continuous math. I think you are using incredibly strong language and in a misguided way.