Number Theory: An Introduction to Mathematics

4 Metric Spaces 31

By generalizing the M ́eray–Cantor method of extending the rational numbers to
the real numbers, Hausdorff (1913) showed that any metric space can be embedded in
a complete metric space. To state his resultprecisely, we introduce some definitions.
A subsetFof a metric spaceEis said to bedenseinEif, for eacha∈Eand each
realε>0, there exists someb∈Fsuch that d(a,b)<ε.
Amapσfrom one metric spaceEto another metric spaceE′is necessarily injec-
tive if it is distance-preserving, i.e. if

d′(σ(a),σ(b))=d(a,b) for alla,b∈E.

If the mapσis also surjective, then it is said to be anisometryand the metric spaces
EandE′are said to beisometric.
A metric spaceE ̄is said to be acompletionof a metric spaceEifE ̄is complete
andEis isometric to a dense subset ofE ̄. It is easily seen that any two completions of
a given metric space are isometric.
Hausdorff’s result says thatany metric space E has a completionE ̄.Wesketchthe
proof. Define two fundamental sequences{an}and{a′n}inEto be equivalent if

lim

n→∞

d(an,a′n)= 0.

It is easily shown that this is indeed an equivalence relation. Moreover, if the funda-
mental sequences{an},{bn}are equivalent to the fundamental sequences{an′},{bn′}
respectively, then

lim

n→∞

d(an,bn)= lim

n→∞

d(an′,b′n).

We can give the setE ̄ of all equivalence classes of fundamental sequences the
structure of a metric space by defining

d ̄({an},{bn})= lim

n→∞

d(an,bn).

For eacha∈E,leta ̄be the equivalence class inE ̄which contains the fundamental
sequence{an}such thatan=afor everyn.Since

d ̄(a ̄,b ̄)=d(a,b) for alla,b∈E,

Eis isometric to the setE′={ ̄a:a∈E}. It is not difficult to show thatE′is dense in
E ̄and thatE ̄is complete.
Which of the previous examples of metricspaces are complete? In example (i), the
completeness ofRnwith respect to the first definition of distance follows directly from
the completeness ofR. It is also complete with respect to the two alternative definitions
of distance, since a sequence which converges with respect to one of the three metrics
also converges with respect to the other two. Indeed it is easily shown that, for every
a∈Rn,

|a|≤|a| 2 ≤|a| 1

and

|a| 1 ≤n^1 /^2 |a| 2 , |a| 2 ≤n^1 /^2 |a|.