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.

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    $\begingroup$ Limits are certainly important, although you usually don't need to go down to the $\epsilon/\delta$ level of limits. Measures are important if you want to prove certain phenomena are general, e.g. that a certain phase exists for a region of parameters and not a measure-zero set of parameters. Closed/open sets are important when dealing with manifolds. Function spaces show up as Hilbert space in QM. $\endgroup$ – Jahan Claes Dec 12 '16 at 20:18
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/27665/2451 and links therein. $\endgroup$ – Qmechanic Dec 12 '16 at 20:25
  • $\begingroup$ @Qmechanic My question is in the context of the day to day work of a research physicist..how much of a role does rigour play, if any? That linked question seems to primarily focus on the historical aspects of mathematical rigour in physics. $\endgroup$ – eurocoder Dec 12 '16 at 20:51
  • $\begingroup$ @JahanClaes I have not worked with manifolds (coming from a numerical mathematics background)..in what way are closed and opens sets applied to physics problems involving manifolds? $\endgroup$ – eurocoder Dec 12 '16 at 20:53
  • $\begingroup$ @eurocoder In GR you might want to know if the universe is an open or closed manifold (or both!), whether it has a boundary, etc. $\endgroup$ – Jahan Claes Dec 12 '16 at 21:16