650 Appendix C • Answers & Hints for Selected Exercises
(a) Let x E R Since A has only finitely many elements, x cannot satisfy
the condition of Thm. 3.2.11. So, x is not a cluster point of A.
(b) A contains all its cluster points, since it doesn't have any.. ·. A is closed.
- Let C denote the collection of all closed sets containing A. Then
(b) VG E c, A~ G so A~ nc =A.
(c) VG EC, nC ~ G. :. A is a subset of every closed subset of A; i.e., A is
the smallest closed set containing A.
( d) A is closed <=? A is the smallest closed set containing A <=? A = A.
(e) The smallest closed set containing 0 [or IR] is 0 [or IR].
- Suppose A =j:. 0 and A is bounded above. By Exercise 7, sup A EAU A'.
- (a) [3,5]U{6} (b), (d), (f), (g) (-oo, l] (c) {1,2,3,4,5,6,7, 8,9} (e) Z
(h), (k) IR (i), (j) {~ : n EN} U {O} (1) Qn[O, l] - x EA<=? x E A or x EA'<=? x E A or every nbd. of x contains a point of
A other than x <=? every nbd. of x contains a point of A. - Suppose A~ B. Then every closed set containing B also contains A. Thus,
A ~ n{ all closed sets containing B} = B. Since B is a closed set containing A ,
A~B. - Let x be a cluster point of A', and c: > 0. Then N"(x) contains a point
a' EA' other than x. Then a' E (x -c:, x ) or (x, x + e). In either case, 30' > 0 3
N 0 (a') ~ N"(x) but x ~ N 0 (a') [draw figure]. Since a' is a cluster point of A ,
N 0 (a') contains a point of A other than a (or x ). Thus, N"(x) contains a point
of A other than x; i.e., x E A'. Therefore, A' contains all its cluster points, and
so is closed. - (a) A^0 is open, so (A^0 )^0 = A^0 • (Thm. 3.1.11)
(b) A is closed, so A= A. (Thm. 3.2.15)
(c) IR - Nxt = A^0 u Ab (by Thm. 3.1. 1 8) =A by Ex. 14.
(d) Ac is closed, so Ace is open. Ac~ Ac, so Ace~ (Ac)c =A. Thus, Ace
is an open subset of A. :. Ace~ A^0 •
A^0 - ~ c A , SO A c ~ A^0 c, which - c is closed. Thus, A c ~ A^0 c = A^0 c. :. A^0 =
Aocc ~Ac. Therefore, A^0 =Ac. - (::::}) x EA::::} x EA or x EA' by Thm. 3.2.17. In the former case, take
{xn} = {x}; in the latter, Thm. 3.2.18 guarantees a sequence in A converging
to x.
( ~) Suppose 3 sequence in A converging to x. Suppose x ~ A. Then 3
sequence of points in A other than x converging to x. Then x E A' by Thm.
3.2.18. Thus, x EA u A'= A. - (::::})Suppose A dense in IR by Defn. 1.5.6. Let r E IR and c: > 0. Then 3
x E A 3 r < x < r + c:, so N"(x) contains a point of A other than r, so r is a
cluster point of A.