Full Space Example

Mathematical Analysis is the branch of mathematical sciences that deals with the study of full space, which is a type of metric space.

A metric space is made up of pairs of points and a function of distance between them; in these spaces it is possible to define a Cauchy sequence that is formed by increasingly smaller distances between these two points. When in the metric space it is no longer possible to find a smaller distance in the sequence then we have a full space. Closed numerical sets, that is, those in which there is a limit, are complete spaces.

Example of Full Space:

The set of natural numbers, including 0, is a complete space since this set is closed by the end of 0. The representation of this number set is N= [0, 1, 2,… n}.

Let's take any two points between two elements of this set, for example 4 and 8, represented in the following way p = (4, 8), the distance function between two points is equal to 4, the Cauchy sequence is given by the sequence {4, 3, 2, 1, 0} that converges on 0.

Another example is the set of positive real numbers formed with {0} which is represented as AND⁺= [0, 1, 2, 3, 4,…. N}, since given two points in this space the Cauchy sequence will converge when the distance is 0

The set of rational numbers is not a complete space, since the distance 0 (the number 0 as a number does not exists in this set) which makes the Cauchy sequence not convergent at any point in this set.

Any closed interval of the natural numbers is a complete space.