Disjoint Open Sets: Spanning Intervals & Uncountable Infinities

[Bob3141592]

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?

 

[CompuChip]

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.

 

[CRGreathouse]

What about this: Can a single open interval be spanned by two or more nontrivial disjoint open subsets of the interval?

 

[CompuChip]

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?

 

[gel]

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.

 

[CRGreathouse]

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.

 

[morphism]

An open interval can't be a nontrivial disjoint union of open sets because it's connected.

 

[gel]

An open interval can't be a nontrivial disjoint union of open sets because it's connected.

of course, that's the simple answer:)