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The sets in T will be called the
T-open
sets of
X
with respect to the topology T. If T is
clear from the context, then we shall simply refer to them as the
open
sets of
X
.
Using induction it is easy to show that condition (3) in the definition of a topol-
ogy can be replaced by
(3¢) The intersection of any
two
sets from T belongs to T.
Definition.
Let
X
be a set. The
discrete topology
on
X
is the set T of
all
subsets of
X
.
Clearly, the discrete topology on a set
X
is defined by the fact that {
p
} is an open
set for each point
p
X
.
Definition.
A
topological space
is a pair (
X
,T) where
X
is a set and T is a topology
on
X
. Again, if T is clear from the context, then one drops the reference to T and
simply says the “topological space”
X
.
Intuitively, when one talks about a topological space, what one is saying is that
one has specified the collection of subsets that will be called “open.” It must be the
case though that the empty set and the whole space are open sets, that any union of
open sets is open, and that the finite intersection of open sets is open.
5.3.1. Example.
If (
X
,d) is a metric space, then (
X
,T) is a topological space, where
T is set of d-open sets. The topology T is called the
topology on
X
induced by d
. In
particular, any subset of
R
n
has a topology induced by the standard Euclidean
metric.
5.3.2. Example.
The Euclidean, taxicab, and max metric on
R
n
induce the same
topology on
R
n
or any of its subsets (Exercise 5.3.1). This topology will be called the
standard topology on
R
n
.
5.3.3. Example.
The topology on
R
n
induced by the discrete metric is the discrete
topology and differs from the standard topology. A less trivial example of distinct
topologies on a space are the topologies on the space of functions C
0
([0,1]) induced
by the metrics d
1
and d
defined by equations (5.2) and (5.3), respectively. For a proof
of this fact see [Eise74].
The following is a useful concept.
Definition.
Let (
X
,T) be a topological space. A
base
for the topology T is a collec-
tion of subsets of
X
, so that each subset belongs to T and every set in T is a union of
elements of the collection.
For example, the open disks
B
n
(
p
,r) in
R
n
are a base for the standard topology on
R
n
. When dealing with open sets of a topology it often suffices to look at elements of
a base. One can also define topologies by means of bases.
