In this exercise we outline a proof of the following theorem: A subset of ℝ is open iff it is the union of countably many disjoint open intervals in ℝ.
(a) Let S be a nonempty open subset of ℝ. For each x ϵ S, let Ax = {a ϵ ℝ: (a, x] ⊆ S} and let Bx = {b ϵ ℝ: [x, b) ⊆ S}. Use the fact that S is open to show that Ax and Bx are both nonempty.
(b) If Ax is bounded below, let ax = inf Ax. Otherwise, let ax = −∞. If Bx is bounded above, let bx = sup Bx; otherwise, let bx = ∞. Show that ax ∉ S and bx ∉ S.
(c) Let Ix be the open interval (ax, bx). Clearly, x ∈ Ix. Show that Ix ⊆ S. (Hint: Consider two cases for y ∈ Ix: y < x and y > x.)
(d) Show that S = ⋃xϵsIx.
(e) Show that the intervals {Ix: x ∈ S} are pairwise disjoint. That is, suppose x, y ∈ S with x ≠ y. If Ix ∩ Iy ≠ ∅, show that Ix = Iy.
(f) Show that the set of distinct intervals {Ix: x ∈ S} is countable.
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