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I will add that even if a published proof is similar to the one in the answer in terms of level of detail, I would still prefer it. My inclination is to trust things that have been published more than answers on MSE, and if there are gaps, I will be more motivated to try to fill them in knowing that the intermediate statements themselves are reliable because they have been made by an expert in that field. This isn't intended as a criticism of MSE or Math Overflow, but I think this is inherent in the nature of a Q&A site.
Also, while I understand that you are curious about my motivation, I wonder why all of this is relevant to determining whether the question is on-topic or not. I don't agree that my motivation ought to make a big difference in terms of the kinds of answers I receive, which should be references to any literature discussing my question or closely related topics.
Quid, I meant the motivation for the mathematical question was self-evident without the sources. I didn't want my question to remain subject to criticism that it was vague. The reason I would like to have this information is closest to (b). I would like to know the proof of this fact in detail. The answers I was given, and I believe any answers to the question on Stack Exchange or Math Overflow, will necessarily omit so many details that they become difficult to verify, for me at least. I am interested in knowing the lemmas and intermediate steps that would be used in a more ordered exposition
I've edited my Stack Exchange question to eliminate the preface, which is not really needed. The motivation for the question is self-evident: the goal is to characterize sets satisfying (1) via a description of their boundary.
I've edited the question. I will point out in my defence that the mathematical content of the question is not vague, and the two answers I received both state that (1) and (2) are indeed equivalent. If those answers are correct, then any vagueness regarding the sources is a moot point because the mathematical premise of my question is correct.
