B U Department of Mathematics
Math 232 Introduction To Complex Analysis
Spring 2008 Exercise Sheet 7
1
In what follows, ”open” and ”closed” means ”open with respect to
C
” and ”closed with respect to
C
”,
respectively;
D
(
z, r
)
stands for the open disc.
1.
Show that arbitrary intersection of open sets is not necessarily open.
2.
Give an example of a subset of
C
which is neither open, nor closed.
3.
Show that the countable union of closed sets is not necessarily closed.
4.
Let
S
=
D
∩
T
where
D
=
D
(0
,
1) and
T
=
{
z
∈
C

Re
z,
Im
z
∈
Q
}
. Show that every point in
S
is a limit point.
5.
Is there a subset
S
of
C
such that
S
has empty interior and
¯
S
=
C
?
6.
Prove that a finite subset of
C
does not have any limit points.
7.
Prove that
a)
a set is open if and only if it contains none of its boundary points.
b)
a set is closed if and only if it contains all of its boundary points.
8.
Let
S
=
{
z
∈
C

0
<
Re
z
≤
1
}
. Is
S
open?
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 Spring '11
 gurel
 Math, Topology, Empty set, Topological space, Closed set, General topology, Rez Imz

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