Together with the set, it makes up a metric space.

Given a metric space a point is called close or near to a set if

In particular, the two conditions are equivalent for metric spaces.

Since the four models describe the same metric space, each can be transformed into the other.

Then one has to work to show that it can be turned to a metric space:

The rational numbers with the same distance also form a metric space, but are not complete.

The positive real numbers with distance function is a complete metric space.

In this case, the two metric spaces are essentially identical.

Complete metric spaces may also fail to have the property.

Therefore only in special cases this distance makes a collection of sets a metric space.

For example a metric space can be regarded as an enriched category.

This real line has several important properties as a metric space:

Let be a metric space (which is not necessarily convex).

On the other hand, if is a compact metric space, then the

The very word' close'only makes sense when the examples are in a metric space, or something like it.

For the particular case of a metric space, this can be expressed as

A set with a metric is called a metric space.

This definition can be extended to the case when f takes values in some metric space.

Also note that any metric space is a uniform space.

L is a complete metric space for all p 1

Conversely, a complete subset of a metric space is closed.

The theorem does not hold as stated for general metric spaces.

Not all metric spaces may be embedded in Euclidean space.

In fact, every compact metric space is a continuous image of the Cantor set.

Distances between points are defined in a metric space.