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Unit 9: The Metric Topology

modern analysis. For example, In this section, we shall define the metric topology and shall give Notes
a number of examples. In the next section, we shall consider some of the properties that metric
topologies satisfy.

9.1 The Metric Topology

9.1.1 Metric Space

Let Xbe any given space.
Let x, y, zX be arbitrary.
A function d : XXR having the properties listed below:
(i) d(x, y)0

(ii) d(x, y) = 0 iff x = y
(iii) d(x, y) = d(y, x)
(iv) d(x, y) + d(y, z)  d(x, z) (triangle inequality)
is called a distance function or a metric for X. Instead of saying, “Let X be a non-empty set with
a metric d defined on it”. We always say, “Let (X, d) be a metric space”.
Evidently, d is a real valued map and d denotes the distance between x and y. A set X, together
with a metric defined on it, is called metric space.

Example 1:
(1) Let X = R and (x, y) = |x – y| x, y  X. Then  is a metric on X. This metric is defined as
usual metric on R.
(2) Let x, yR be arbitrary

ì 0 iff x = y
Let (x, y) = í
î 1 iff x  y
Then  is a metric on R.
This metric is defined as trivial metric or discrete metric on R.

9.1.2 Pseudo Metric Space

Let X   be any given space. Let x, y, zX be arbitrary. A function d : XXR having the
properties listed below:

(i) d(x, y)0,
(ii) d(x, y) = 0 if x = y,
(iii) d(x, y) = d(y, x),
(iv) d(x, y) + d(y, z)  d(x, z),
Where x, y, zX

is called pseudo metric on x. The set X together with the pseudo metric d is called pseudo metric
space. Pseudo metric differs from metric in the sense that.
d(x, y) = 0 even if xy

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