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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 R^{1} 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 R^{1} 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
Apostol's definition is that the component interval of an open subset S of R^{1} 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 R^{1} 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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