metric space

Complete metric spaces may also fail to have the property.

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

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

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

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

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

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