Chapter 1
Metric spaces
The theory of metric spaces is called “metric topology”.
Definition 1.1. A metric space is a pair (X,d), where X is a set and d is a function called a
metric, such that d:XXR and
(a)
(x,yX) d(x,y)=0  x=y
(b)
(x,yX) d(x,y) = d(y,x)
(c)
(x,y,zX) d(x,y)+d(x,z)d(x,z) (triangle inequality)
The concept of a metric generalizes the notion of the distance between points in a
space. On a plane or in a 3D space we usually measure the distance as the length of the
segment connecting the points. This is not always the best way. If you are in a city, the
distance you actually have to cover in order to get from A to B is not the length of the
segment AB but much more. But the three conditions of the definition are what all sensible
ways of distance measuring have in common.
Proposition 1.1. For every two points x and y in a metric space (X,d), d(x,y)0.
Proof. d(x,y) = ½(d(x,y) + d(y,x))  ½(d(x,x) = 0. 
Example 1.1. (Rn,dE), where dE denotes the Euclidean metric, d((x1,x2,…,xn), (y1,y2,…,yn))=
n
 (x
i 1
i
 yi ) 2 . This is the “normal” metric, the length of the segment connecting the two
points. In particular, for n=1 we get dE(x,y) =
( x  y ) 2 = |xy|.
Example 1.2. (R2,dC), where dC((a,b),(x,y)) = |ax| + |by|. This is known as the “city
metric”, as this is the distance you have to travel from (a,b) to (x,y) if the streets in the city are
perpendicular. The first two conditions of the definition of the metric are trivially fulfilled. To
verify the third let us consider dC((a,b),(x,y)) + dC((x,y),(p,q)) = |ax| + |by| + |xp| + |yq| =
|ax| + |xp| +|by| + |yq|  |ap| + |bq| = dC((a,b),(p,q)).
Example 1.3. (R2,dRC), where dRC((a,b),(x,y)) = dE((a,b),(x,y)), if the line passing through the
points (a,b) and (x,y) contains the origin (0,0), or dRC((a,b),(x,y)) = dE((a,b),(0,0)) +
dE((0,0),(x,y)) , if it does not. This metric is known as the “railroad center metric” and is
modeled on a country in which all roads radiate from a single point ( the “origin”). Strange as
it seems it is a metric nevertheless. The first two conditions are trivially satisfied. To prove
the third we must consider two cases. First, at least two of the three points involved belong to
a line passing through the origin. Then the distances between the points are the same as if they
all belonged to one line through the origin and the condition follows from the fact that (R,dE)
is a metric space. The second case is that no two of the three point A, B and C lie on a
common radius. Then dRC(A,B) + dRC(B,C) = dE(A,0) + dE(0,B) + dE(B,0) + dE(0,C) =
dRC(A,C) + 2dE(0,B)  dRC(A,C).
Example 1.4. (X,dD), where dD(x,y) = 1 whenever xy and dD(x,x) = 0. Again the first two
conditions are easy. Consider any x,y,z from X. If x=z then dD(x,z) = 0, so the inequality
holds. If xz then dD(x,z)=1 and dD(x,y)+ dD(y,z)1, because y cannot be equal to both x and
z. This metric is called the discrete metric. Notice that the nature of elements of X is of no
importance. We can define the discrete metric on any set whatsoever.
Example 1.5. If (X,d) is a metric space then every subset A of X is a metric space with
respect to d restricted to AA.
Many definitions known from calculus can be easily transformed into much more
general context of a metric space. One example is the limit of a sequence.
Definition 1.2. Let (xn) be a sequence of points of a metric space (X,d) and L be a point from
X. Then
lim x n  L iff (>0)(k)(n>k) d(xn,L)<
n
Notice that the only difference between this definition and the one used in calculus is
that here we use d(xn,L) instead of |xnL|. In other words the “normal” convergence is a
special case of the general concept of convergence, used in the metric space (R,dE). This
general concept of convergence, of course, that the terms of the sequence are arbitrarily close
to the limit if subscripts are large enough. It is the meaning of the word “close” that varies
from one metric space to another. Sequences that have limits are called convergent, those who
don’t – divergent.
Proposition 1.2. lim x n  L iff lim d ( x n , L)  0
n
n 
Proof. lim d ( x n , L)  0 iff (>0)(k)(n>k) |d(xn,L)0|<. Since d(xn,L)0, the last
n 
inequality is identical with d(xn,L)< and the two conditions are identical.
Theorem 1.1. lim x n  L1 and lim x n  L2 then L1=L2. In other words, a convergent
n 
n 
sequence has exactly one limit.
Proof. Let =½d(L1,L2). Suppose >0. Then, by Definition 1.2, there exists k such that
d(xn,L1)< and d(xn,L2)< for each n>k. By the triangle inequality we get d(L1,L2)
d(xn,L1)+d(xn,L2) < 2 = d(L1,L2). Since this is impossible, we have  = ½d(L1,L2) = 0, hence
d(L1,L2)=0 and L1 = L2.
Theorem 1.2. If a sequence (xn) is convergent then (>0)(k)(n,p>k) d(xn,xp)<.
Proof. Suppose lim x n  L . Then (1>0)(k)(n>k) d(xn,L)<1. If we choose 1=½ we get
n
for any n,p>k, d(xn,xp)  d(xn,L) + d(xp,L)<21=.
The condition (>0)(k)(n,p>k) d(xn,xp)< is known as the Cauchy condition and
sequences satisfying it are called Cauchy sequences. The last theorem states that every
convergent sequence is a Cauchy sequence. The opposite is not always true as the sequence
( 1n ) is not convergent in ((0,1),dE).
Example 1.6. The sequence (1, 1n ) is convergent to (1,0) in (R2,dE) and is divergent in
(R2,dRC). Indeed, dE((1, 1n ),(1,0)) = 1n and lim
n 
1
n
 0 . On the other hand, for every n and p, the
line through (1, 1n ) and (1, 1p ) does not pass through the origin, so dRC((1, 1n ),(1, 1p )) =
dE((1, 1n ),(0,0)) + dE((1, 1p ),(0,0)) > 2, so (1, 1n ) is not a Cauchy sequence.
Example 1.7. Every sequence convergent in (R2,dRC) is also convergent in (R2,dE) and the
limits are the same. This follows easily from the fact that for every to points on the plane, the
Euclidean distance is not greater then the railroad-center distance.
Example 1.8. A sequence (xn) is convergent to L in a discrete metric space iff there exists k,
such that for all n>k, xn=L. This is self evident.
Some geometrical notions can be easily generalized to metric spaces (circle, disc,
ball). Others, that are more specific to “real” spaces are hard to define in a metric space.
Among these are line, angle, direction and so on.
Definition 1.3. Let (X,d) be a metric space, xX and rR. A ball centered at a point x and
with the radius r is the set B(x,r) = {yX:d(x,y)<r}. A circle is the set C(x,r) = {yX : d(x,y)
= r}.
Using the concept of a ball we can redefine the limit of a sequence.
Proposition 1.3. lim x n  L iff (>0)(k)(n>k) xnB(L,)
n