An infinite union of closed sets?

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SUMMARY

The discussion centers on the properties of infinite unions of closed sets in topology. It establishes that while individual closed sets can form unions that are either open, closed, or neither, specific examples illustrate these outcomes. For instance, the union of sets defined as Cn = [1/n, 1-1/n] results in an open interval (0, 1), while Cn = [-n, n] leads to the entire set of real numbers, which is both open and closed. The conversation emphasizes the importance of definitions in topology, particularly regarding boundary points and neighborhoods.

PREREQUISITES
  • Understanding of basic topology concepts, including open and closed sets.
  • Familiarity with the definitions of boundary points and neighborhoods.
  • Knowledge of real number intervals and their properties.
  • Ability to analyze the implications of infinite unions in set theory.
NEXT STEPS
  • Study the definitions and properties of open and closed sets in topology.
  • Learn about boundary points and their significance in set theory.
  • Explore examples of infinite unions and intersections of sets in various contexts.
  • Investigate the implications of the empty set in topology and its classifications.
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Mathematicians, students of topology, and anyone interested in the properties of sets and their unions in real analysis.

  • #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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