Graphics Reference
In-Depth Information
5.3.4. Theorem.
Let
X
be a set and W a collection of subsets of
X
satisfying:
(1) Each element of
X
belongs to some subset of W.
(2) If
O
1
,
O
2
W and
x
O
1
«
O
2
, then there exists an
O
W with
x
O
O
1
«
O
2
.
Then there is a unique topology T on
X
for which W is a base.
Proof.
See [Eise74].
Definition.
Let (
X
,T) be a topological space. A subset
A
of
X
is called
T-closed
(or
simply
closed
if T is clear from the context), if
X
-
A
is T-open.
5.3.5. Theorem.
Let (
X
,T) be a topological space. Then
(1) Both the empty set f and the whole space
X
are closed sets.
(2) An arbitrary intersection of closed subsets of
X
is closed.
(3) Any finite union of closed subsets of
X
is closed.
Proof.
This is easy. See the proof of Theorem 5.2.4.
Definition.
X
. A subset
V
of
X
that con-
tains
p
is called a
neighborhood of
p
if there is a open set
U
so that
p
Let
X
be a topological space and let
p
V
. A col-
lection of neighborhoods of
p
is called a
local base at
p
or a
neighborhood base at
p
if
every neighborhood of
p
contains a member of this collection.
U
Definition.
A topological space (
X
,T) is said to be
metrizable
if
X
admits a metric d
so that T is the topology on
X
induced by d.
Not all topological space are metrizable.
5.3.6. Example.
Let
X
= {
p
,
q
} and T = {f,
X
, {
p
}, {
p
,
q
}}. Then (
X
,T) is a non-
metrizable topological space. If it were metrizable, then
p
and
q
would have disjoint
neighborhoods, which is not the case here.
For less trivial examples of nonmetrizable spaces see [Eise74]. Any space that can
be “drawn” (and whose topology is reflected by the picture) will be metrizable because
it lives in
R
n
.
Definition.
A topological space is said to be a
Hausdorff
space if any two distinct
points of
X
have disjoint neighborhoods.
The Hausdorff separability condition may seem like an obvious condition that
should always hold but it does not follow from our definition of a topology. It is
certainly satisfied for metrizable spaces and it is satisfied by all the topological
spaces we shall look at, but, as we just saw in Example 5.3.6, not every topological
space is a Hausdorff space. Most of the time, however, it is an important technical
condition that is assumed for some results to hold. We shall run into it later in quite
a few places.
