I don't know whether many mathematical objects are defined to defy intuition per se, but counterintuitive results often occur through an effort to prove a state

I don't know whether many mathematical objects are defined to defy intuition per se, but counterintuitive results often occur through an effort to prove a statement along the lines of "Not all objects X have property Y" (see the sibling comment's counterexamples in topology). One nice example of "pathological" behaviour came out of Cantor's study of the continuum [0]: he gave an example of a function that is continuous and has zero derivative almost everywhere (i.e. everywhere except at a set of isolated points), but that manages to increase from 0 to 1. https://en.wikipedia.org/wiki/Cantor_function https://en.wikipedia.org/wiki/Cantor_function