# Is discrete metric space complete?

## Is discrete metric space complete?

In a space with the discrete metric, the only Cauchy sequences are those which are constant from some point on. Hence any discrete metric space is complete.

## How can you prove the completeness of a metric space?

A metric space (X, ϱ) is said to be complete if every Cauchy sequence (xn) in (X, ϱ) converges to a limit α ∈ X. There are incomplete metric spaces. If a metric space (X, ϱ) is not complete then it has Cauchy sequences that do not converge. This means, in a sense, that there are gaps (or missing elements) in X.

Why is discrete metric complete?

So in discrete metric space, every Cauchy sequence is constant sequence and that way every Cauchy sequence is convergent sequence. Thus we conclude the discrete metric space is complete.

### Is discrete metric space continuous?

That is, any function from a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discrete metric space to another metric space bounded by 1 is short. is constant.

### Which metric space is complete?

The space R of real numbers and the space C of complex numbers (with the metric given by the absolute value) are complete, and so is Euclidean space Rn, with the usual distance metric.

What is a discrete metric space?

metric space any set of points, the discrete metric specifies that the distance from a point to itself equal 0 while the distance between any two distinct points equal 1.

#### What makes a metric space complete?

We say that a metric space (X, d) is complete if every Cauchy sequence in X has a limit in X, i.e., every Cauchy sequence is convergent. 1.4 Example. Rn with the Euclidean metric is complete.

#### What is the completion of a metric space?

Definition. A completion of a metric space (X, d) is a pair consisting of a complete metric space (X∗,d∗) and an isometry ϕ: X → X∗ such that ϕ[X] is dense in X∗. Theorem 1. Every metric space has a completion.

What is meant by discrete metric space?

## Why is discrete metric not compact?

Notice that any subset of a metric space with the discrete metric is closed and bounded. However, only finite subsets are compact (by a homework question), hence any infinite subset is closed, bounded, and not compact.

## What is usual metric space?

A metric space is a set X together with a function d (called a metric or “distance function”) which assigns a real number d ( x , y ) to every pair x , y such that X satisfies the properties (or axioms ):

What is metric and metric space?

In mathematics, a metric space is a non-empty set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points.

### How do you prove that every metric space is discrete?

Every metric space contains a discrete, coarsely dense subset 1 Proving that a subset endowed with the discrete metric is both open and closed – choice of radius of the ball around a point 1 Proving a set is both open and closed 5 Prove that every infinite metric space \$(X, d)\$ contains an infinite subset \$A\$ such that \$(A, d)\$ is discrete. 1

### What is a discrete space in math?

a) a metric space that happens to induce the discrete topology (i.e., where each singleton set is open, such as in your example) and d ( x, y) = { 0 x = y 1 x ≠ y.

Is every subset of a discrete metric space open?

Proving that a subset endowed with the discrete metric is both open and closed – choice of radius of the ball around a point Related 4 Show that for a finite metric space A, every subset is open 2 Every metric space contains a discrete, coarsely dense subset 1

#### What are the properties of the metric space of a set?

Your very example is excellent to remind us that some properties of metric spaces (such as completeness) are really properties of the metric space, not of the induced topology: The set { 1 n: n ∈ N } endowed with the metric inherited from R is discrete, but is not a discrete metric space; and it is not complete with respect to this metric.