What Is Space Completion?

Convergence is defined in any metric (and, more generally, topological) space X. A sequence {ak}, 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 C(a, r) = {x: d(a, x) < r} with center a and radius r, we say that x is the limit of a sequence {xk} and write x = limk→∞xk, if, for any r > 0, there is an index K such that d(x, xk) < r, for all k > K.

By the triangle inequality, d(x, xk) < r and d(x, xm) < r imply d(xm, xk) < 2r, meaning that the elements of a convergent sequence grow closer to each other. A not necessarily convergent sequence that has this property is known as Cauchy sequence. Spaces in which every Cauchy sequence is convergent, i.e., has a limit, are said to be comlete.

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 d(r, s) = |r - s|. Some sequence of rational numbers converge, i.e., converge to a rational number. Such is, for example, the sequence {1/k} that converges to 0. But some sequences of rational numbers, albeit Cauchy, do not converge, i.e., do not converge to a rational number. Such is, for example, {10-1 + 10-4 + ... + 10-k2}. Now, where do we go from here? Cauchy sequences of rational numbers may not converge to a rational number but they are all known to converge to real numbers. In a more general case, where do the "real" numbers come from? Any real number is a limit of a sequence of rational numbers. For example, it's a limit of the sequence of its abbreviated decimal expansions. In addition, every number on the number line can be approached from left as well as from right; the approximation may oscillate from left to right and back; the approximation can approached the limit slowly or fast. In short, the same real number is a limit of great many sequences of rationals. In so far as we talk about reals, all such sequences lead to the same result, i.e., the same real number. And here lies the key to handling a more general case.

Definition

Two Cauchy sequences {ak} and {bk} are said to be equivalent if, for any r > 0, there is an index K such that, for all k > K, d(ak, bk) < r.


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({ak}, {bk}) = lim d(ak, bk).

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:

      {ak} + {bk} = {ak + bk}
    {ak} × {bk} = {ak × bk}.

    and then check that these operations possess all the expected properties.

  • The elements of X may be identified with constant sequences: qk = 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 d(r, s) = |r - s|. In general, however, a different notion of proximity or metric may or may not lead to the same completion of a topological or metric space. For Q, there is a metric that leads to a completion different from R. We'll look into this elsewhere.

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