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.