3.2 Closed Sets and Cluster Points 153- Find the closure of each of the following sets:
(a) (3, 5) U {6}
(c) {1,2,3,4,5,6, 7,8,9}
(e) Z
(g) (-oo, 0) U [O, l]
(i) { ~ : n EN}
(k) Q(b) (-oo, 0) U (0, 1)
(d) (-oo,O)U[O,l)
(f) (-oo, 0) U (0, l]
(h) JR!. - {l, 2, 3}
(j) { ~ : n E N} U {O}
(1) Q n (O, 1)
Prove that x EA iff every neighborhood of x contains a point of A.
Prove that Ab= AnAc, which yields an alternate proof that Ab is closed.
Prove that if As;;: B, then As;;: B.
Suppose A,B s;;: R Prove that AU B =AUE and An B s;;: AnB. Show
by example that A n B and A n B are not necessarily equal.
Prove that \IA s;;: JR!., the set A' of all its cluster points is closed.
Prove that the set of cluster points of a bounded sequence (see Definition
2.6.14) is a closed set.
Prove the following identities:
(a) Aoo =Ao
(d)Acclc=Ao
(b) A cl cl = A cl
(e) Ab= N^1 - A^0( c) Nl = JR!. - Nxt
(f) Acl o cl o = Acl o- Prove that x E A is an isolated point of A iff it is not a cluster point of
A. - Prove that x EA iff 3 sequence of points of A converging to x.
- Prove the sequential criterion for open sets: A set A is open iff ~
sequence in Ac converging to a point in A. Use this criterion to prove that
the interval (0, 1) is open, but the interval [O, 1) is not open. - Prove that A is dense in JR!. (in the sense of Definition 1.5.6) iff every real
number is a cluster point of A. - Prove that A is dense in JR!. iff A= R [See Exercise 3.1.20.]
- Given A, B s;;: JR!., prove that A is dense in B if and only if every neigh-
borhood of every point of B contains a point of A. (Equivalently, every
open set containing a point of B contains a point of A.) - Prove Theorem 3.2.21.