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then the open ball
B
n
(
p
,e) is contained in all the sets
O
i
and hence in their intersec-
tion, proving (2). Parts (3) and (4) follow from parts (1) and (2) and the identity
k
k
(
)
=-
=
I
U
n
n
RO R
-
O
.
i
i
i
=
1
i
1
The identities
•
•
11
1
1
Ê
Ë
ˆ
¯
È
Í
˘
I
U
˙
=
()
-
,
=
0
and
,
1
-
01
,
nn
nn
n
=
1
n
=
2
show that arbitrary intersections of open sets need not be open and arbitrary unions
of closed sets need not be closed.
Definition.
A point
p
is a
limit
or
accumulation point
of a set
X
if every neighbor-
hood of
p
meets
X
in a point other than
p
. More precisely, for every e>0,
(
)
«πf
n
)
-
{}
(
Bp
,
e
p
X
.
An
isolated point
of
X
is a point of
X
that has a neighborhood containing no point of
X
other than itself (that is, it is not a limit point).
For example, 1 is a limit point of (0,1), but 1.0001 is not because
B
1
(1.0001,0.00005) is disjoint from (0,1). A point
p
in the set
X
is
not
necessarily a
limit point of
X
. Only those points whose neighborhoods contain infinitely many
points of
X
are. For example, if
X
= [0,1] » {2}, then 2 is an isolated point and
not
a
limit point of
X
, but every point in [0,1]
is
a limit point of
X
.
Definition.
The
closure
of a set
X
, denoted by cl(
X
), is defined by
()
=»
{
}
cl
XX pp
is a limit point of
X
For example,
(
()
)
=
[]
(
[]
)
=
[]
cl
01
,
01
,
and
cl
01
,
01
, .
The closure of the set of rationals in the reals is the reals.
4.2.2. Proposition
(1) For every set
X
, cl(
X
) is a closed set.
(2) If
X
is closed, then cl(
X
) =
X
.
Proof.
We prove (1) and leave (2) as an exercise (Exercise 4.2.4). We need to show
that
=-
()
n
YR
cl
X
