An infinite union of closed sets?

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Discussion Overview

The discussion revolves around the properties of infinite unions of closed sets, specifically whether such unions can be classified as closed, open, or neither. Participants explore various examples and definitions related to closed and open sets within the context of topology.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants argue that if each set in the union is closed, the union can still be closed, as in the case of C0 = [0, 1].
  • Others present counterexamples where the union of closed sets results in an open set, such as Cn = [1/n, 1 - 1/n], leading to the union being (0, 1).
  • Another example discussed is Cn = [1/n, 1], where the union results in (0, 1], which is neither open nor closed.
  • One participant suggests that the union of Cn = [-n, n] for all positive integers n results in the set of all real numbers, which they claim is both open and closed.
  • There is a discussion about the definition of openness and closedness, with some participants emphasizing the need to show that the complement of the union is open to classify the union as closed.
  • Several participants highlight that definitions of closed sets can vary, including the definitions based on limit points and boundary points.
  • Concerns are raised about the implications of discussing properties of sets that do not exist, with references to philosophical considerations in mathematics.

Areas of Agreement / Disagreement

Participants express differing views on the classification of infinite unions of closed sets, with no consensus reached. Some examples lead to agreement on certain properties, while others remain contested.

Contextual Notes

Participants note that definitions of open and closed sets can vary, and the discussion includes unresolved mathematical steps and assumptions regarding the nature of unions and complements.

  • #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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