I know very little in the way of math history, but I question that was bothering me recently is where the terms open and closed came from in topology. I know that it's easy to ascribe a sense of openness/closedness to said sets, but I feel like there are a lot of other, more appropriate words that could have been used. On a related note, I was wondering who first developed the idea of open/closed sets, and who first used those words to describe them.
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This answer addresses your question on who first actually used the words "open" and "closed" to describe open and closed sets. It seems, according to this article, like the first mention of this language was in René Baire's doctoral dissertation,
which I believe first appears in page 7 of the document in the context of defining an "open domain." The term itself was first actually defined by Lebesgue in his dissertation, Intégrale, longueur, aire, (which I cannot find an online copy of) for the purpose of setting up Lebesgue measure.
I think the rest of your question should be addressed to your satisfaction between the linked article and related MSE/MO threads.
the followings archives corresponds to the doctoral thesis of Henri Lebesgue:
You can search for Analysis situs, old name for the actual topology.Lebesgue and Rham (french mathematicians) wrote about that subject.
Hugo Mancera Colombia