Disjoint Open Sets: Spanning Intervals & Uncountable Infinities

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Discussion Overview

The discussion revolves around the properties of disjoint open sets in relation to spanning continuous intervals, particularly focusing on whether the union of such sets can cover an entire interval. The scope includes theoretical considerations in topology and the implications of connectedness in open intervals.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant questions whether the union of disjoint open sets can span a continuous interval, even if the collection is uncountably infinite.
  • Another participant asserts that the union cannot span a closed interval, referencing topological properties, and notes that open intervals can be spanned by a single disjoint open set.
  • A participant inquires if a single open interval can be represented by two or more nontrivial disjoint open subsets, suggesting a potential misunderstanding of the term "span."
  • One participant challenges the notion of triviality in the answer, suggesting that the problem reduces to considering closed bounded intervals, which are compact.
  • Another participant emphasizes that an open interval cannot be a nontrivial disjoint union of open sets due to its connectedness.

Areas of Agreement / Disagreement

Participants express differing views on the implications of spanning intervals with disjoint open sets, with some asserting that it is impossible due to connectedness, while others explore the nuances of the term "spanning." There is no consensus on the interpretation of "spanning" or the implications of uncountable collections of open sets.

Contextual Notes

Participants note that the discussion hinges on the definitions of "spanning" and the properties of connectedness in topology. The implications of closed versus open intervals are also highlighted, but the discussion does not resolve these complexities.

Bob3141592
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Am I correct in thinking that the union of disjoint open sets cannot span a continuous interval? Assume that each of the sets is a proper subset of the interval. Does this apply even if the collection of open sets is uncountable infinite?
 
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You are correct in thinking that, as long as the interval is closed of course. In fact, you cannot even do it with non-disjoint sets. Otherwise, it would mean that the set [0, 1] is open in R (see the properties of a topology).

For an open interval, it is trivial, e.g. ]0, 1[ is spanned by a single disjoint open set.
 
What about this: Can a single open interval be spanned by two or more nontrivial disjoint open subsets of the interval?
 
CRGreathouse, the answer seems to be quite trivially "no". So probably I am missing here, and my guess the problem is in the word "span".

What exactly is meant by "spanning" in this context?
 
I wouldn't call it trivial.
Still, not too difficult - note that the interval between any two disjoint open intervals is a closed interval. So the problem reduces to the case of a closed bounded interval, which is compact.
 
CompuChip said:
CRGreathouse, the answer seems to be quite trivially "no". So probably I am missing here, and my guess the problem is in the word "span".

What exactly is meant by "spanning" in this context?

First of all, I have no interest in the answer -- I just thought this may have been the question intended (though not written!) by the OP.

But I simply meant for the union of the subsets to be the full set. I agree that this appears trivially impossible.
 
An open interval can't be a nontrivial disjoint union of open sets because it's connected.
 
morphism said:
An open interval can't be a nontrivial disjoint union of open sets because it's connected.

of course, that's the simple answer:)
 

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