If you delete the integers from [1,infinity) you are left with the sets (0,1), (1,2), (2,3) ..... and you are left with the trivial observation that the union of a collection of disjoint open sets is the union of the same collection of disjoint open sets.
Personally, I vote for Kolmogorov. I think the whole point is that an open set can be expressed as the union of disjoint sets not necessarily open (on the real line). Even Taylor in his theorem does not refer specifically to E1, E2... as open sets even though conditions (i)-(iii) do. That's what makes his Theorem so maddening.
Sorry, I added an edit after your response:
"I think the whole point is that an open set can be expressed as the union of disjoint sets not necessarily open (on the real line). Even Taylor in his theorem does not refer specifically to E1, E2... as open sets even though conditions (i)-(iii) do. That's what makes his Theorem so maddening."
OK, to be even more specific, I think Taylor is wrong and youi don't. That's probably as good a place to end it as any.
You are right, I missed that. Technically, I copied it from Taylors book. OK, I don't understand Taylor's version.
For reference I include Kolmogorov's version again which makes sense to me (note reference below takes you directly to the theorem, you don't have to hunt for it):
http://books.google.com/books?id=OyW...20line&f=false
Shilov: "Every open set on the real line is a finite or countable union of nonintersecting open intervals."
Rudin: "Every open set in R1 is the union of an at most countable collection of disjoint segments."
Its still a tie.
By the way, you can get almost any used book cheap at abe.com (hope its OK to mention this. If not, say so and I will edit it out).
EDIT: Just as a reminder: (0,1) and [1,2) are disjoint (non-intersecting).
Shilov (clearly) and Taylor (obtuseley) state:
Every open set S on the real line is a countable union of nonintersecting open intervals.
Example:
S=
period.
Kolmogorov and Rudin state:
Every open set S on the real line is (can be expressed as) a countable union of nonintersecting intervals.
Example:
S=
or
S=
or.......
The examples make everything clear. Shilov's proof became crystal clear once I understood what the theorem was saying. Taylor and Klomogorov's proofs were'nt clear, and Rudin's was an excercise.
What confused me all along was that I thiought Taylor was saying, for example,
If S=(1,3) it can be expressed as