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Why are unions, in particular, infinite unions, of open sets open allowed by the definition of topology? Such a condition allows for interiors of sets to always exist, but why does this matter?

user109871
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  • Because it reflects how we think of open sets in metric spaces from analysis, perhaps? I don't know which one came first, though. In the end, however, the real answer is that that's the definition that turned out to work. – Arthur Nov 17 '17 at 16:44
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    In the last minutes you are posting so many questions- which one should we start with? See here. – Dietrich Burde Nov 17 '17 at 16:45
  • If you run a search on this site you will find that this question already has many good answers here. This is only an example. – Giuseppe Negro Nov 17 '17 at 16:45

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Because in metric spaces, the union of infinitely many open sets is open.

Duncan Ramage
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    The question to you now is, are you certain that the concept of "open" in metric spaces came before topology was a subject? Otherwise, this isn't an answer, but could just as well be pointing out that analysis found useful terminology in topology and borrowed it. – Arthur Nov 17 '17 at 16:45
  • @Arthur well, even if the concept of open sets in metric spaces did not precede this specific development of topology, I think Duncan's answer still gives one reason why we would want the sets in a topology to satisfy this property. – Theoretical Economist Nov 17 '17 at 16:58
  • @Arthur Cauchy died in 1857 and Weierstrass in 1897, meanwhile, Poincaré published Analysis Situs in 1895. This isn't iron clad of course, since lots of topological ideas predate Poincaré, and I do not know if any of the people who foundationalized calculus explicitly referred to "open sets", but it's consistent and plausible. – Duncan Ramage Nov 17 '17 at 19:03