# 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
• $\{\in\mathbb R^2 |x^2+y^2=9\}$ (circle with radius 3, centred at origin)
• $\{\in \mathbb R^3 |x^2+y^2+z^2=4\}$ (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.