### 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= 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 of M such that diam < ϵ (k=1,2,…,n) and such that . (Wikipedia)**Cover of a set**: If for a set A, if it is a subset of union of sets then we say 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 {} 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 is bounded iff it is contained in a square of finite side length.
- A subset of is bounded iff is contained in a cube of finite side length.
- In a set is totally bounded if and only if it is bounded. Hence, any bounded subset of is an example of totally bounded subset of
- is totally bounded if and only if it is bounded A subset A of is totally bounded if and only if A contains only finite number of points.

### Examples and Counterexamples

- In , [0,∞) is not bounded. But in , 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 , let be the sequence whose all terms are 0 except kth term. Consider the set . Then E is bounded with diameter √2

**Bounded and totally bounded**: [0,2] in R- (circle with radius 3, centred at origin)
- (sphere with radius 2, centred at origin)
- Finite sets in

**Bounded but not totally bounded**: Interval [0,2] in

## 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

- Bounded Sets on Wikipedia
- Totally Bounded Sets on Wikipedia
- Bounded Sets on WolframAlpha
- Bounded Sets on LibreTexts
- Bounded Sets on MathOnline
- Which metric spaces are totally bounded? on Math.SE

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

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