An infinite union of closed sets?

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The discussion revolves around the properties of infinite unions of closed sets, highlighting that such unions can be open, closed, or neither, depending on the specific sets involved. Examples include C0 = [0, 1], where the union remains closed, and Cn = [1/n, 1-1/n], where the union results in an open interval (0, 1). The conversation also explores the case where Cn = [-n, n], leading to the conclusion that the union is the entire set of real numbers, which is both open and closed. Participants debate definitions of open and closed sets, emphasizing that the empty set is considered both open and closed. Ultimately, the discussion illustrates the complexities and nuances of set theory in topology.
  • #31
Er, Hurkyl, maybe you can tell me if that was Ok?
 
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  • #32
pivoxa15 said:
I have just realized something 'illogical' about the definition of open sets according to 'R contains none of its boundary points'. Surely 'its boundary points' refer to R's boundary points so R has boundary points. But than to state R contains none of its boundary points is confusing.
You do understand, I hope, that "A is closed if it contains all of its boundary points" is not intended to be a strict, logical definition. It is shorthand for "If the statement (if x is a boundary point of A the x is contained in A) is true then A is closed". For A= R, the statement "if x is a boundary point of R then x is contained in R" is true for all R because the premises are false.

If every statement had to be in rigorously true logic, then every mathematics book would be as long (and as hard to read) as "Principia Mathematica"!
 

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