This is because the natural account of the negation of a proposition in such a space says that it holds on the largest open set contained in the Boolean complement of the set of points on which the original proposition held, which is in general smaller than the Boolean complement. However, specifying a topological space by its closed sets is every bit as reasonable as specifying it by its open sets. Yet the logic of closed sets is known to be paraconsistent, ie. This is because the natural account of the negation of a proposition, namely that it holds on the smallest closed set containing the Boolean negation of the proposition, means that on the overlapping boundary both the proposition and its negation hold. Thus dynamical theories determine their own logic of possible propositions, and corresponding theories which may be inconsistent, and are certainly as natural as their incomplete counterparts.
Category theory throws light on many mathematical structures.
It has certainly been proposed as an alternative foundation for mathematics. Such generality inevitably runs into problems similar to those of comprehension in set theory, see eg. Hence there is the same possible application of inconsistent solutions. There is also an important collection of categorial structures, the toposes, which support open set logic in exact parallel to the way sets support Boolean logic. This has been taken by many to be a vindication of the foundational point of view of mathematical intuitionism.