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3.12: More on Cluster Points and Closed Sets. Density

This page is a draft and is under active development. 

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I. The notions of cluster point and closed set (§§12, 14) can be characterized in terms of convergent sequences. We start with cluster points.

Theorem 3.12.1

(i) A sequence {xm}(S,ρ) clusters at a point pS iff it has a subsequence {xmn} converging to p.

(ii) A set A(S,ρ) clusters at pS 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=limnxmn, 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

xm1Gp(1).

Next, choose a term

xm2Gp(12) with m2>m1.

(Such terms exist since Gp(12) contains infinitely many xm.) Next, fix

xm3Gp(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)xmnGp(1n), i.e., ρ(xmn,p)<1n0,

whence ρ(xmn,p)0, or xmnp. (Why?) Thus we have found a subsequence xmnp, 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.

Example 3.12.1

(a) Recall that the set R of all rationals clusters at each pE1 (§§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=11 and x2n1=00.

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.

Theorem 3.12.1 (Bolzano-Weierstrass).

(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

azmb,

we have

ainfzmpsupzmb

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,ymE1.

If {¯zm} is bounded, all ¯zm are in some square [¯a,¯b]. (Why?) Let

¯a=(a1,a2) and ¯b=(b1,b2).

Then

a1xmb1 and a2ymb2 in E1.

Thus by the first part of the proof, {xm} has a convergent subsequence

xmkp1 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 xmp1, and a2ymb2, as before.

We now reapply this process to {ym} and obtain a subsubsequence

ymip2 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 AEn(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 ¯zmA. 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.

Definition

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=AA.

Theorem 3.12.1

We have p¯A in (S,ρ) iff each globe Gp(δ) about p meets A, i. e.,

(δ>0)AGp(δ).

Equivalently, p¯A iff

p=limnxn 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).

Theorem 3.12.1

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., AA.

(ii) A=¯A.

(iii) A contains the limit of each convergent sequence {xn}A (if any).

Proof

Parts (i) and (ii) are equivalent since

AAA=AA=¯A.(Explain!)

Now let A be closed. If pA, then pA; therefore, by Definition 3 in §12, some Gp fails to meet A(GpA=). Hence no pA 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., pA but noGp is entirely in A. This means that each Gp meets A. Thus

p¯A (by Theorem 3),

and

p=limnxn for some {xn}A (by the same theorem),

even though pA( for pA).

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. AB¯A¯B.

Corollary 3. ¯A is always a closed set A.

Corollary 4. ¯AB=¯A¯B (the closure of AB 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)=(pδ,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).

Definition

Given AB(S,ρ), we say that A is dense in B iff each globe Gp pB, meets A. By Theorem 3, this means that each pB is in ¯A; i.e.,

p=limnxn for some {xn}A.

Equivalently, AB¯A.3.

Summing up, we have the following:

A is open iff A=A0.

A is closed iff A=¯A; equivalently, iff AA.

A is dense in B iff AB¯A.

A is perfect iff A=A.


This page titled 3.12: More on Cluster Points and Closed Sets. Density is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Elias Zakon (The Trilla Group (support by Saylor Foundation)) via source content that was edited to the style and standards of the LibreTexts platform.

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