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June 17, 2004
MATH 215 Homework 2
Turn in by June 22, 2004 until 9:40 a.m.
Each question is 4 points, total: 20 points.
1.
For the following sets
E
ﬁnd
E
0
and int(
E
). Are these sets open, closed, neither?
a) In (
R
,
 · 
),
E
=
{
(

1)
n
+ (1
/m
) :
n,m
∈
N
}
b) In (
R
2
,d
2
),
E
=
{
(
x,y
) : 1
≤
x
2
+
y
2
<
4
}
.
Solution.
a) Since (

1)
n
is either +1 or

1, the set
E
=
{
1 +
1
1
,

1 +
1
2
,

1 +
1
3
,

1 +
1
4
,...
} ∪ {
1 +
1
1
,
1 +
1
2
,
1 +
1
3
,
1 +
1
4
,...
}
.
Then
E
0
=
{
1
,
1
}
, int(
E
) =
∅
.
E
is neither open nor closed.
b)
E
is an annulus (i.e. the region between two concentric circles where the inner boundary
is included, but the outer boundary is excluded).
E
0
=
{
(
x,y
) : 1
≤
x
2
+
y
2
≤
4
}
,
int(
E
) =
{
(
x,y
) : 1
< x
2
+
y
2
<
4
}
.
E
is neither open nor closed.
2.
Prove that in a metric space the union of an arbitrary collection of open sets is open.
Proof.
Let
{
G
α
:
α
∈
A
}
be an arbitrary (ﬁnite, countable or uncountable) collection of
open subsets of
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This note was uploaded on 04/15/2010 for the course INDUSTRIAL ie513 taught by Professor Zeynephuygur during the Spring '10 term at Bilkent University.
 Spring '10
 zeynephuygur

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