Are mathematical notions like closed sets, limits of sequences, measures, and function spaces basically irrelevant in the day to day work of a physicist? Naturally, such concepts are the foundations upon which everything stands in both mathematics and physics, but do physicists need to concern themselves with the fine details of these concepts in their daily work or can they 'get away with' not having to use them while performing research?
Particularly in the areas of solid state physics, quantum mechanics, relativity, optics, and electromagnetism. Do any/some/all of these fields regularly/sometimes/never have to go that level of mathematical rigour? If these concepts do get used regularly in some particular fields, it would be great to hear some examples of how they arise and why they are necessary.