Page 53 - DMTH401_REAL ANALYSIS

Unit 3: Matric Spaces

Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following Notes
generalization of the triangle inequality:

d(x, S)  d(x, y) + d(y, S)
which in particular shows that the map is continuous.
Given two subsets S and T of M, we define their Hausdorff distance to be

d (S, T) = max {sup {d(s, T) : s  S}, sup {d(t, S) : t  T}}
H
In general, the Hausdorff distance d (S, T) can be infinite. Two sets are close to each other in the
H
Hausdorff distance if every element of either set is close to some element of the other set.
The Hausdorff distance d turns the set K(M) of all non-empty compact subsets of M into a metric
H
space. One can show that K(M) is complete if M is complete. (A different notion of convergence
of compact subsets is given by the Kuratowski convergence.)
One can then define the Gromov–Hausdorff distance between any two metric spaces by
considering the minimal Hausdorff distance of isometrically embedded versions of the two
spaces. Using this distance, the set of all (isometry classes of) compact metric spaces becomes a
metric space in its own right.

3.1.3 Product Metric Spaces

If (M , d ), ....., (M , d ) are metric spaces, and N is the Euclidean norm on R , then (M1 x .. x Mn,
n
1 1 n n
N(d1, ..., dn)) is a metric space, where the product metric is defined by
N(d1, ..., dn) ((x1, ...., xn), (y1, . . ., yn)) = N(d1(x1, y1), . . ., dn(xn, dn)),
and the induced topology agrees with the product topology. By the equivalence of norms in
finite dimensions, an equivalent metric is obtained if N is the taxicab norm, a p-norm, the max
norm, or any other norm which is non-decreasing as the coordinates of a positive n-tuple
increase (yielding the triangle inequality).
Similarly, a countable product of metric spaces can be obtained using the following metric

 1 di(x , y )
i
i
d(x, y) = å .
i
i 1 2 1 d (x , y )+
=
i i i
R
An uncountable product of metric spaces need not be metrizable. For example, R is not first-
countable and thus isn’t metrizable.
Continuity of Distance
+
It is worth noting that in the case of a single space (M, d), the distance map d: M  M  R (from
the definition) is uniformly continuous with respect to any of the above product metrics N(d, d),
and in particular is continuous with respect to the product topology of M  M.

Quotient Metric Spaces

If M is a metric space with metric d, and ~ is an equivalence relation on M, then we can endow the
quotient set M/~ with the following (pseudo)metric. Given two equivalence classes [x] and [y],
we define
d’([x], [y]) = inf {d(p , q ) + d(p , q ) + . . . + d(p , q )
1 1 2 2 n n

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