In this exercise we outline a proof of the following theorem: A subset of is open iff it is the union of countably many disjoint open intervals in . (a) Let S be a nonempty open subset of . For each...

In this exercise we outline a proof of the following theorem: A subset of

is open iff it is the union of countably many disjoint open intervals in
.

(a) Let S be a nonempty open subset of
. For each x ∈ S, let A_x
= {a ∈

: (a, x ] ⊆ S } and let B_x
= {b ∈ R : [x, b) ⊆ S }. Use the fact that S is open to show that A_x
and B_x
are both nonempty.

(b) If A_x
is bounded below, let a_x
= inf A_x. Otherwise, let a_x
= −
. If B_x
is bounded above, let b_x
= sup B_x; otherwise, let b_x
=
. Show that a_x
∉ S and b_x
∉ S.

(c) Let
_Ix
be the open interval (a_x
, b_x). Clearly, x ∈ I x. Show that I x ⊆ S. (Hint: Consider two cases for y ∈ I x : y <> x.)

(d) Show that S = ∪<sub>x
∈

S</sub>
I_x.

(e) Show that the intervals {I_x
: x ∈ S} are pairwise disjoint. That is, suppose x, y ∈ S with x ≠ y. If I_x
∩ I_y
≠ ∅, show that I_x
= I_y.

(f) Show that the set of distinct intervals {I x : x ∈ S} is countable.