What does the term complementary interval mean?

[AxiomOfChoice]

What does the term "complementary interval" mean?

I'm studying the Cantor set and related notions right now, and I keep coming across the term "complementary interval." What does that mean? I can't find a definition spelled out anywhere!

 

[Hurkyl]

Where are you seeing this? How is it being used?

Without knowing any context, I would guess that "complementary interval" is not a technical term -- it's simply an English phrase built from the two technical terms "complement" and "interval", or maybe even from the English word "complementary" and the technical term "interval".

 

[AxiomOfChoice]

Ok. Allow me to provide some context: "The difficult part is verifying continuity of the Cantor function at points of the Cantor set that have a sequence of complementary intervals converging to them."

 

[JG89]

I know nothing about what you are studying, but a wild guess: could it mean that the intervals don't overlap? If I and I' are any two intervals, then they are complimentary intervals if there exists no points in I which are in I' ?

 

[Hurkyl]

The complement of the Cantor set in [0,1] is a countable union of disjoint open intervals. Maybe they're talking about those?

 

[AxiomOfChoice]

The complement of the Cantor set in [0,1] is a countable union of disjoint open intervals. Maybe they're talking about those?

Yes, that's a reasonable idea. But how do you make sense of intervals NOT in the Cantor set converging to points that ARE in the Cantor set? What does it even MEAN for a sequence of open intervals to converge to a point?

 

[Hurkyl]

Well, the most straightforward idea I can think of is:

A sequence of sets S_n converges to a point P iff, for every open set U containing P, there exists an N such that n>N implies S_n is a subset of U.

Or equivalently, every sequence s_n of points chosen so that s_n is an element of S_n converges to P.