3.12: More on Cluster Points and Closed Sets. Density
This page is a draft and is under active development.
( \newcommand{\kernel}{\mathrm{null}\,}\)
I. The notions of cluster point and closed set (§§12, 14) can be characterized in terms of convergent sequences. We start with cluster points.
(i) A sequence {xm}⊆(S,ρ) clusters at a point p∈S iff it has a subsequence {xmn} converging to p.
(ii) A set A⊆(S,ρ) clusters at p∈S iff p is the limit of some sequence {xn} of points of A other than p; if so, the terms xn can be made distinct.
- Proof
-
(i) If p=limn→∞xmn, then by definition each globe about p contains all but finitely many xmn, hence infinitely many xm. Thus p is a cluster point.
Conversely, if so, consider in particular the globes
Gp(1n),n=1,2,…
By assumption, Gp(1) contains some xm. Thus fix
xm1∈Gp(1).
Next, choose a term
xm2∈Gp(12) with m2>m1.
(Such terms exist since Gp(12) contains infinitely many xm.) Next, fix
xm3∈Gp(13), with m3>m2>m1,
and so on.
Thus, step by step (inductively), select a sequence of subscripts
m1<m2<⋯<mn<⋯
that determines a subsequence (see Chapter 1, §8) such that
(∀n)xmn∈Gp(1n), i.e., ρ(xmn,p)<1n→0,
whence ρ(xmn,p)→0, or xmn→p. (Why?) Thus we have found a subsequence xmn→p, and assertion (i) is proved.
Assertion (ii) is proved quite similarly - proceed as in the proof of Corollary 6 in §§14; the inequalities m1<m2<⋯ are not needed here. ◻
(a) Recall that the set R of all rationals clusters at each p∈E1 (§§14, Example (e)). Thus by Theorem 1(ii), each real p is the limit of a sequence of rationals. See also Problem 6 of §§12 for ¯p in En.
(b) The sequence
0,1,0,1,…
has two convergent subsequences,
x2n=1→1 and x2n−1=0→0.
Thus by Theorem 1(i), it clusters at 0 and 1.
Interpret Example (f) and Problem 10(a) in §14 similarly.
As we know, even infinite sets may have no cluster points (take N in E1). However, a bounded infinite set or sequence in En (*or Cn) must cluster. This important theorem (due to Bolzano and Weierstrass) is proved next.
(i) Each bounded infinite set or sequence A in En (* or Cn) has at least one cluster point ¯p there (possibly outside A.
(ii) Thus each bounded sequence in En (* or Cn) has a convergent subsequence.
- Proof
-
Take first a bounded sequence {zm}⊆[a,b] in E1. Let
p=¯limzm.
By Theorem 2(i) of Chapter 2, §13, {zm} clusters at p. Moreover, as
a≤zm≤b,
we have
a≤infzm≤p≤supzm≤b
by Corollary 1 of Chapter 2, §13. Thus
p∈[a,b]⊆E1,
and so {zm} clusters in E1.
Assertion (ii) now follows - for E1− by Theorem 1(i) above.
Next, take
{¯zm}⊆E2,¯zm=(xm,ym);xm,ym∈E1.
If {¯zm} is bounded, all ¯zm are in some square [¯a,¯b]. (Why?) Let
¯a=(a1,a2) and ¯b=(b1,b2).
Then
a1≤xm≤b1 and a2≤ym≤b2 in E1.
Thus by the first part of the proof, {xm} has a convergent subsequence
xmk→p1 for some p1∈[a1,b1].
For simplicity, we henceforth write xm for xmk,ym for ymk, and ¯zm for ¯zmk. Thus ¯zm=(xm,ym) is now a subsequence, with xm→p1, and a2≤ym≤b2, as before.
We now reapply this process to {ym} and obtain a subsubsequence
ymi→p2 for some p2∈[a2,b2].
The corresponding terms xmi still tend to p1 by Corollary 3 of §14. Thus we have a subsequence
¯zmi=(xmi,ymi)→(p1,p2) in E2
by Theorem 2 in §15. Hence ¯p=(p1,p2) is a cluster point of {¯zm}. Note that ¯p∈[¯a,¯b] (see above). This proves the theorem for sequences in E2 (hence in C).
The proof for En is similar; one only has to take subsequences n times. (*The same applies to Cn with real components replaced by complex ones.)
Now take a bounded infinite set A⊂En(∗Cn). Select from it an infinite sequence {¯zm} of distinct points (see Chapter 1, §9, Problem 5). By what was shown above, {¯zm} clusters at some point ¯p, so each G¯p contains infinitely many distinct points ¯zm∈A. Thus by definition, A clusters at ¯p.◻
Note 1. We have also proved that if {¯zm}⊆[¯a,¯b]⊂En, then {¯zm} has a cluster point in [¯a,¯b]. (This applies to closed intervals only.)
Note 2. The theorem may fail in spaces other than En(∗Cn). For example, in a discrete space, all sets are bounded, but no set can cluster.
II. Cluster points are closely related to the following notion.
The closure of a set A⊆(S,ρ), denoted ¯A, is the union of A and the set of all cluster points of A call it A′. Thus ¯A=A∪A′.
We have p∈¯A in (S,ρ) iff each globe Gp(δ) about p meets A, i. e.,
(∀δ>0)A∩Gp(δ)≠∅.
Equivalently, p∈¯A iff
p=limn→∞xn for some {xn}⊆A.
- Proof
-
The proof is as in Corollary 6 of §14 and Theorem 1. (Here, however, the xn need not be distinct or different from p. ) The details are left to the reader.
This also yields the following new characterization of closed sets (cf. §12).
A set A⊆(S,ρ) is closed iff one of the following conditions holds.
(i) A contains all its cluster points (or has none); i.e., A⊇A′.
(ii) A=¯A.
(iii) A contains the limit of each convergent sequence {xn}⊆A (if any).
- Proof
-
Parts (i) and (ii) are equivalent since
A⊇A′⟺A=A∪A′=¯A.(Explain!)
Now let A be closed. If p∉A, then p∈−A; therefore, by Definition 3 in §12, some Gp fails to meet A(Gp∩A=∅). Hence no p∈−A is a cluster point, or the limit of a sequence {xn}⊆A. (This would contradict Definitions 1 and 2 of §14.) Consequently, all such cluster points and limits must be in A, as claimed.
Conversely, suppose A is not closed, so −A is not open. Then −A has a noninterior point p; i.e., p∈−A but noGp is entirely in −A. This means that each Gp meets A. Thus
p∈¯A (by Theorem 3),
and
p=limn→∞xn for some {xn}⊆A (by the same theorem),
even though p∉A( for p∈−A).
We see that (iii) and (ii), hence also (i), fail if A is not closed and hold if A is closed. (See the first part of the proof.) Thus the theorem is proved. ◻
Corollary 1. ¯∅=∅.
Corollary 2. A⊆B⟹¯A⊆¯B.
Corollary 3. ¯A is always a closed set ⊇A.
Corollary 4. ¯A∪B=¯A∪¯B (the closure of A∪B equals the union of ¯A and ¯B).
III. As we know, the rationals are dense in E1 (Theorem 3 of Chapter 2, §10). This means that every globe Gp(δ)=(p−δ,p+δ) in E1 contains rationals. Similarly (see Problem 6 in §12), the set Rn of all rational points is dense in En. We now generalize this idea for arbitrary sets in a metric space (S,ρ).
Given A⊆B⊆(S,ρ), we say that A is dense in B iff each globe Gp p∈B, meets A. By Theorem 3, this means that each p∈B is in ¯A; i.e.,
p=limn→∞xn for some {xn}⊆A.
Equivalently, A⊆B⊆¯A.3.
Summing up, we have the following:
A is open iff A=A0.
A is closed iff A=¯A; equivalently, iff A⊇A′.
A is dense in B iff A⊆B⊆¯A.
A is perfect iff A=A′.