174 Chapter 3 • Topology of the Real Number System
- \:/n E .N, let Ln denote the set of all left endpoints of the disjoint closed
intervals comprising Cn· For example, L1 = {O, n and £2 = {o, ~' ~, n.
(a) Show that \:/n E .N, Ln consists of those numbers in [O, 1] having a
terminating base-3 decimal of no more than n digits using only O's
and 2's. [Use mathematical induction.]
(X)
(b) Let L = LJ Ln. Prove that Lis dense in C.
n=l- Prove that 0, IR, (-oo,a], [a,+oo), and [a,b], are perfect sets, for any
real numbers a, b with a < b. - Prove that a set of real numbers is perfect iff it is closed and has no
isolated points. - Prove Theorem 3.4.17.
- Prove that if A is nowhere dense, then Ac is dense in IR. Give a coun-
terexample to show that the converse is not true. - Prove that A is closed, then A is nowhere dense iff Ac is dense in IR.
13. That C has measure zero may seem intuitively consistent with the fact
that C is nowhere dense. Show by an example that a set can be dense in
IR and still have measure zero.- Use mathematical induction to prove that if A 1 , A 2 , · · · , An are measur-
able, then A 1 U A 2 U · · · U An is measurable, andμ (Q
1Ai) :::; n~l μ(Ai)·
- Show that (μ5) follows from (μ3) and (μ4), and that (μ6) follows from
(μ1) and (μ5). - Show that the ( =?) direction of (μ9) follows from (μl), (μ6), and (μ7).
- Prove Corollary 3.4.22.
18. (Project) "Open Middle nth" Cantor-like Sets: Repeat the con-
struction of the Cantor set given in Definition 3.4.1, but at each stage
remove the "open middle fourth" (or fifth, or... ). Prove that the re-
sulting Cantor-like set has measure zero. [Hint: Don't bother with the
actual intervals comprising each Cn; only μ( Cn) is significant.] Which of
Theorems 3.4.2 and 3.4.3, Lemma 3.4.4, and Theorems 3.4.12, 3.4.15, and
3.4.17 remain true for the resulting set?