3.1 Neighborhoods and Open Sets 145- Suppose A is a bounded, nonempty set of real numbers. Prove that sup A
and inf A are boundary points of A. Also prove that if A is open then
sup A rf. A and inf A rf. A. - Find all the isolated points of the sets given in Exercise 3.1.2.
- In Example 3.1.20 we asserted that Q has no isolated points. Prove that
assertion. Does the set of all irrational numbers have any isolated points?
Justify your answer. - Prove Theorem 3.1.21.
- Prove Theorem 3.1.22.
- (a) Prove that (An B)^0 = A^0 n B^0 •
(b) Prove that A^0 UB^0 s:;; (AUB)^0.
(c) Give an example of sets A and B such that (AU B)^0 of. A^0 U B^0 •
Prove that A^0 =A - Ab.
Prove that a set A s:;; JR is dense in JR (see Definition 1.5.6) iff Vx E JR,
every neighborhood of x contains a point of A.
Prove that a set A s:;; JR is dense in JR iff every nonempty open set of real
numbers contains a point of A.
Prove that a sequence {xn} converges to a real number Liff every open
set containing L contains all but finitely many terms of {xn}·
Exercises 22 and 23 are for students who have studied "countable" sets,
as in Section 2.8.- Prove that every nonempty open set A is the union of countably many
open intervals with rational endpoints. [Hint: Consider intervals of the
form ( r - ~, r + ~) , where r E A n Q and n E N.] - Prove that every nonempty open set A is the union of countably many
pairwise disjoint open intervals. [Suggestion: Define C = {Ia : a E A},
where Ia= U{all open subintervals of A containing a}. Prove that each
Ia is an open interval containing a. Then prove that the Ia's are pairwise
disjoint and A = U C. Finally, prove that C is a countable collection.]