Analysis/Topology: Bounded and Totally Bounded Sets

Definitions

  • Bounded set: Let <M,ρ> be a metric space. We say that the subset A of M is bounded if there exists a positive number L such that ρ(x,y)≤L  (x,y∈A) (Wikipedia)
  • Diameter of a set: If A is bounded, we define diameter of A (denoted by diam A) as diam A={l.u.b.}_(x,y\in A)  ρ(x,y). If A is not bounded we write diam A=∞. (Wikipedia)
  • Totally bounded sets: Let <M,ρ> be a metric space. The subset A of M is said to be totally bounded if given ϵ>0, there exist a finite number of subsets A_1 , A_2 , ..., A_n of M such that diam A_k < ϵ (k=1,2,…,n) and such that A\subset \cup_{k=1}^n A_k(Wikipedia)
  • Cover of a set: If for a set A, if it is a subset of union of sets A_1,A_2,... then we say A_k cover A. (Wikipedia)
  • Totally bounded sets: A⊂M is totally bounded iff for every ϵ>0, A can be covered by a finite number of subsets of M whose diameters are all less than ϵ.
  • ϵ-dense: Let A be a subset of the metric space M. The subset B of A is said to be ϵ-dense in A (where ϵ>0) if for every x∈A there exists y∈B such that ρ(x,y)<ϵ. That is, B is ϵ-dense in A if each point of A is within distance ϵ from some point of B.

Theorems

  • If the subset A of the metric space <M, ρ> is totally bounded, then A is bounded.
  • The subset A of the metric space <M, ρ> is totally bounded if and only if for every ϵ>0, A contains a finite subset {x_1,x_2,…,x_n} which is ϵ-dense in A.
  • Let <M,ρ> be a metric space. The subset A of M is totally bounded if and only if every sequence of points of A contains a Cauchy subsequence.
  • A subset of \mathbb R^2 is bounded iff it is contained in a square of finite side length.
  • A subset of \mathbb R^3 is bounded iff is contained in a cube of finite side length.
  • In \mathbb R a set is totally bounded if and only if it is bounded. Hence, any bounded subset of \mathbb R is an example of totally bounded subset of \mathbb R
  • \mathbb R^n is totally bounded if and only if it is bounded A subset A of \mathbb R_d is totally bounded if and only if A contains only finite number of points.

Examples and Counterexamples

  • In \mathbb R, [0,∞) is not bounded. But in \mathbb R_d, ρ(x,y)≤1,∀x,y∈[0,∞) and hence it is bounded. (L=1). Hence, all subsets of $latex \mathbb R_d$ are bounded.
  • In \mathscr{l}^2, let e_k be the sequence whose all terms are 0 except kth term. Consider the set E= \cup_{n=1}^\infty {e_k }. Then E is bounded with diameter √2
  • Bounded and totally bounded: [0,2] in R
    • \{<x,y>\in\mathbb R^2 |x^2+y^2=9\} (circle with radius 3, centred at origin)
    • \{<x,y,z>\in \mathbb R^3 |x^2+y^2+z^2=4\} (sphere with radius 2, centred at origin)
    • Finite sets in \mathbb R_d
  • Bounded but not totally bounded: Interval [0,2] in \mathbb R_d

Quizzes

1. Which of the following set is totally bounded in
[0,∞)
A set is bounded iff it is totally bounded in
[0,1)
(-∞, 2)
A set is bounded iff it is totally bounded in
All of the above
A set is bounded iff it is totally bounded in . [0,1) is the only bounded set in
Let M be a metric space. Which of the following is true?
A totally bounded set in a metric space need not be bounded.
A set is totally bounded then it is bounded.
A bounded set in a metric space need not be totally bounded.
Interval [0,2] in
If M = , then every set is bounded and totally bounded.
(-∞, 2) is a subset of and is neither bounded nor totally bounded.
If M = , then every set is bounded and totally bounded.
[0,2] in is bounded but not totally bounded.

External Reading Materials

This article was previously posted in my old blog – reposted here with additional insights.

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