Apostol definition of component interval

  • #1
Apostol in his "Mathematica Analysis" defines something called a "component interval". However, I cannot find it anywhere on google or in other books I have on analysis, and I really would like to see a picture of what he means..

Apostol's definition is that the component interval of an open subset S of R1 is an open interval I such that I[itex]\subseteq[/itex]S and such that no open interval J≠I exists such that I[itex]\subseteq[/itex]J[itex]\subseteq[/itex]S

In other words, a component interval of S is not a proper subset of any other open interval in S.

So does this mean basically that if we cut up R1 into disjoint open intervals and define their union as S, then a component interval I will be the any one of those disjoint open intervals such that I spans the whole of one such disjoint open interval?

I attached to this post an image i drew in Paint of how I visually see component intervals. If someone could please look on it and tell me if i am correct i would be grateful
 

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  • #2
lavinia
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Apostol in his "Mathematica Analysis" defines something called a "component interval". However, I cannot find it anywhere on google or in other books I have on analysis, and I really would like to see a picture of what he means..

Apostol's definition is that the component interval of an open subset S of R1 is an open interval I such that I[itex]\subseteq[/itex]S and such that no open interval J≠I exists such that I[itex]\subseteq[/itex]J[itex]\subseteq[/itex]S

In other words, a component interval of S is not a proper subset of any other open interval in S.

So does this mean basically that if we cut up R1 into disjoint open intervals and define their union as S, then a component interval I will be the any one of those disjoint open intervals such that I spans the whole of one such disjoint open interval?

I attached to this post an image i drew in Paint of how I visually see component intervals. If someone could please look on it and tell me if i am correct i would be grateful
I would think about whether a if you had two component intervals in a set whether they can intersect or not. This should give you a picture of what they are.

Some sets have only an empty component interval - like a single point or the rational numbers or the irrational numbers.
 
  • #3
Bacle2
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This may have to see with the result that every open subset of ℝ can be expressed uniquely as the disjoint union of countably-many open intervals (e.g., for countability, select a rational for each interval). A component interval may be one of the intervals in the decomposition of the set. Are you reading Luke, Mark, etc?
 
  • #4
This may have to see with the result that every open subset of ℝ can be expressed uniquely as the disjoint union of countably-many open intervals (e.g., for countability, select a rational for each interval). A component interval may be one of the intervals in the decomposition of the set. Are you reading Luke, Mark, etc?
ok i think i get it then. And no I havent heard of Luke or Mark, I am reading Apostol's "Mathematical Analysis" and Rudin's "Principles of Mathematical Analysis". Do Luke, Mark also have good analysis books? Could you please recommend?
 
  • #5
Bacle2
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Sorry, Kawashay, it was a stupid joke about apostols (of which I really know nothing). I personally like M. Rosenlicht's book and Wilcox and Myers' Intro to Lebesgue Integration and Fourier series for intro/review books.

BTW: this result about open sets simplifies a lot of definitions about measure.
 
  • #6
Sorry, Kawashay, it was a stupid joke about apostols (of which I really know nothing). I personally like M. Rosenlicht's book and Wilcox and Myers' Intro to Lebesgue Integration and Fourier series for intro/review books.

BTW: this result about open sets simplifies a lot of definitions about measure.
Ohh you meant those religious guys ahah. Sorry i am not christian or whoever believes in them lol so i didnt get joke. But thanks I will try to find those books you mentioned, and what do you mean about this result simplifying definitions about measure?
 
  • #7
Bacle2
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I learnt all the names by watching Jeopardy; I want to play one day, make it big, and retire and do math and hang-out all day .

Anyway, enough daydreaming: one of the ways this helps is in the definition of outer measure m*of a set, which is defined (sort of) recursively: the outer measure m*(a,b):=b-a,
and m* of any subset (notice _every_ set has a well-defined outer-measure) is defined as the inf m* over all covers by open sets. Since open sets have a unique decomposition, this allows us well-define, e.g., the measure of any open set as the disjoint union of open intervals as Ʃ(an-bn) , since the decomposition into open intervals is unique.
 
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