Number Theory: An Introduction to Mathematics

4 Metric Spaces 27

4 Metric Spaces

The notion of convergence is meaningful not only for points on a line, but also for
points in space, where there is no natural relation of order. We now reformulate our
previous definition, so as to make it more generally applicable.
Theabsolute value|a|of a real numberais defined by

|a|=a ifa≥ 0 ,

|a|=−a ifa< 0.

It is easily seen that absolute values have the following properties:

| 0 |= 0 ,|a|>0ifa= 0 ;

|a|=|−a|;

|a+b|≤|a|+|b|.

The first two properties follow at once from the definition. To prove the third, we ob-
serve first thata+b≤|a|+|b|,sincea≤|a|andb≤|b|. Replacingaby−aandb
by−b, we obtain also−(a+b)≤|a|+|b|.But|a+b|is eithera+bor−(a+b).
Thedistancebetween two real numbersaandbis defined to be the real number

d(a,b)=|a−b|.

From the preceding properties of absolute values we obtain theircounterparts for dis-
tances:

(D1)d(a,a)= 0 ,d(a,b)> 0 if a=b;

(D2)d(a,b)=d(b,a);

(D3)d(a,b)≤d(a,c)+d(c,b).

The third property is known as thetriangle inequality, since it may be interpreted as
saying that, in any triangle, the length of one side does not exceed the sum of the
lengths of the other two.
Fr ́echet (1906) recognized these three properties as the essential characteristics of
any measure of distance and introduced the following general concept. A setEis a
metric spaceif with each ordered pair (a,b)ofelementsofEthere is associated a real
number d(a,b), so that the properties(D1)–(D3)hold for alla,b,c∈E.
We note first some simple consequences of these properties. For alla,b,a′,b′∈E
we have

|d(a,b)−d(a′,b′)|≤d(a,a′)+d(b,b′)(∗)

since, by(D2)and(D3),

d(a,b)≤d(a,a′)+d(a′,b′)+d(b,b′),

d(a′,b′)≤d(a,a′)+d(a,b)+d(b,b′).

Takingb=b′in(∗), we obtain from(D1),

|d(a,b)−d(a′,b)|≤d(a,a′). (∗∗)