Lecture 1
1
Review
of Topology
1.1
Metric
Spaces
Definition 1.1.
Let
X
be
a set. Define the Cartesian
product
X
×
X
=
{
(
x, y
) :
x, y
∈
X
}
.
Definition 1.2.
Let
d
:
X
×
X
R
be a
mapping. The mapping
d
is
a
metric on
→
X
if the following four
conditions
hold
for all
x, y, z
∈
X
:
(i)
d
(
x, y
) =
d
(
y, x
),
(ii)
d
(
x, y
)
≥
0,
(iii)
d
(
x, y
) = 0
⇐⇒
x
=
y
,
and
(iv)
d
(
x, z
)
≤
d
(
x, y
) +
d
(
y, z
).
Given
a metric
d
on
X
,
the
pair
(
X, d
)
is
called
a
metric space
.
Suppose
d
is
a metric
on
X
and
that
Y
⊆
X
. Then
there is
an
automatic metric
d
Y
on
Y
defined by restricting
d
to
the subspace
Y
×
Y
,
d
Y
=
d Y
×
Y.
(1.1)

Together
with
Y
,
the
metric
d
Y
defines
the automatic metric space (
Y, d
Y
).
1.2
Open and Closed Sets
In
this section we
review
some
basic definitions
and
propositions
in
topology.
We
review open sets,
closed sets,
norms, continuity, and
closure. Throughout
this
section,
we
let
(
X, d
)
be
a metric
space
unless
otherwise specified.
One
of
the
basic
notions
of
topology
is
that
of the
open
set. To
define an
open
set, we
first
define
the
�
neighborhood.
Definition 1.3.
Given a point
x
o
∈
X
, and
a
real number
� >
0, we define
U
(
x
o
, �
) =
{
x
∈
X
:
d
(
x, x
o
)
< �
}
.
(1.2)
We
call
U
(
x
o
, �
)
the
�
neighborhood of
x
o
in
X
.
Given a subset
Y
⊆
X
,
the
�
neighborhood
of
x
o
in
Y
is
just
U
(
x
o
, �
)
∩
Y
.
Definition 1.4.
A
subset
U
of
X
is
open
if for every
x
o
U
there exists
a
real
∈
number
� >
0 such that
U
(
x
o
, �
)
⊆
U
.
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 Fall '04
 unknown
 Topology, Metric space, Topological space, open sets, XO, point xo

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