1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 119

Definition 2.44 A complete metric space S* is said to be a completion

of the space S iff (1) Sis a subspace of S*, and (2) Sis everywhere dense

in S, i.e., S = S.

Example The space JR of the real numbers is the completion of the space

Q of the rationals.

Definition 2.45 A one-to-one mapping f : S __, S' of a metric space

(S,d) onto a metric space (S',d') is said to be isometric if, for arbitrary

points x 1 , x 2 E S, the equality

holds. In such a case the spaces themselves are said to be isometric.

Theorem 2.31 (Completion of a metric space). Every incomplete metric
space has a completion and all its completions are isometric.

Outline of Proof The idea of the proof is similar to that involved in the
Cantor-Meray theory of real numbers (see, e.g., references [10] and [15]).
Two Cauchy sequences {xn.}, {x~} in the metric space (S, d) are said
to be equivalent if limn.-+oo d(xn, x~) = 0. This is an equivalence relation,
since it is clearly reflexive, symmetric, and transitive. From this it follows
that all Cauchy sequences in S can be partitioned into equivalent classes
of sequences. Each of these classes is then considered as an element of a
new space S*. Any sequence in a class serves as a representative for the

whole class. A metric d is introduced in S in the following manner: If

x and y are any two members of S, we choose representative sequences
{xn.} and {yn} for x and y*, respectively, and set

d*(x*,y*) = n-+oo lim d(xn.,Yn)

Then it is shown that this definition of distance in S is independent of
the particular representatives chosen for the classes x and y, and that it
satisfies all the metric axioms. Next, it is shown that S can be embedded
isometrically in S by letting the class x of Cauchy sequences converging
to a point x E S correspond to that point x, and that S is then everywhere
dense in S. Finally, it is proved that the space S is complete, and that
any other completion of S is necessarily isometric to S.
In matters depending only on the distance between elements ( conver-
gence, completeness, etc.) isometric spaces are considered as equal. For
the details of the proof, the reader may consult [4], [13], or [14].