Apostol definition of component interval

[kahwawashay1]

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 anyone 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

 

[lavinia]

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 anyone 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.

 

[Bacle2]

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?

 

[kahwawashay1]

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 haven't 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?

 

[Bacle2]

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.

 

[kahwawashay1]

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?

 

[Bacle2]

I learned 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(a_n-b_n) , since the decomposition into open intervals is unique.