In 1952 Grothendieck proved a result connecting the question of when countably compact subspaces of certain function spaces are compact with the ability to interchange double limits, as is often done in Analysis. Iovino and colleagues connected the interchange of double limits to questions of definability of pathological Banach spaces. In his recent lecture in this seminar, he connected that interchange to questions in Machine Learning. With my recent Ph.D. student, Clovis Hamel, I extended Iovino’s work to deal with such definability in not necessarily compact logics. Previously I had answered questions of Arhangel’skii concerning generalizations of Grothendieck’s work by showing they were undecidable. Today I will speak about the topology involved in both of these endeavours.