What Is Space Completion?
Convergence is defined in any metric (and, more generally, topological) space X. A sequence {a_{k}}, k = 1, 2, 3, ... is a set of elements indexed by natural numbers. A generalized sequence may be indexed by an ordered set in which, for any pair of indices, there is an index exceeding both indices in the pair. A (generalized) sequence is convergent, if there is an element in X, called the limit of the sequence, such that, starting with some index, the elements of the sequence fall into a prescribed neighborhood of the limit.
In a metric space X with metric d where neighborhoods may be defined as balls
By the triangle inequality,
Completion of a space is an expansion that includes the limits of all Cauchy sequences, including the ones that do not converge. Does not this sound strange? To include the non-existent limits! To understand how is this possible let's look at an example.
Consider the set Q of rational numbers with usual metric
Definition
Two Cauchy sequences {a_{k}} and {b_{k}} are said to be equivalent if, for any
Instead of the original space, we shall consider all the Cauchy sequences in this space. In a metric space, we may define a distance function between two sequence:
d({a_{k}}, {b_{k}}) = lim d(a_{k}, b_{k}). |
One needs to prove of course that the so defined function is indeed a metric. In fact, it is not, because the distance between equivalent Cauchy sequences is 0. We'll have to identify, i.e., declare "the same" all equivalent sequences to make the function a metric.
One needs to do more; actually a good deal more. I'll describe a few steps that have to be taken.
The equivalence of Cauchy sequences is indeed an equivalence relation, meaning that all Cauchy sequences fall into equivalence classes, with distance 0 between two sequences only if they belong to the same class.
It is possible to define arithmetic operations on sequences. The most natural way is to do that componentwise:
{a_{k}} + {b_{k}} = {a_{k} + b_{k}}
{a_{k}} × {b_{k}} = {a_{k} × b_{k}}.and then check that these operations possess all the expected properties.
The elements of X may be identified with constant sequences:
q_{k} = q, for all k. Constant sequences are obviously Cauchy. This allows for embedding of the rationals into the space of equivalence classes of the rational Cauchy sequences.The resulting space Y is complete. The space itself and the process of which it was the end result, both are called the completion of X. However, completeness of Y is not obvious; it must be proven because the Cauchy sequences in Y are not quite the same as the Cauchy sequences in X.
The original space X is dense in Y, meaning that in every neighborhood of a element y∈Y there is an element x∈X.
The set of all reals R is the completion of the rationals Q. More accurately, R is the completion of Q when Q is endowed with the common metric
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