# 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.