27th January 2005
Munkres
§
23
Ex. 23.1.
Any separation
X
=
U
∪
V
of (
X,
T
) is also a separation of (
X,
T
). This means that
(
X,
T
) is disconnected
⇒
(
X,
T
) is disconnected
or, equivalently,
(
X,
T
) is connected
⇒
(
X,
T
) is disconnected
when
T
⊃ T
.
Ex. 23.2.
Using induction and [1, Thm 23.3] we see that
A
(
n
) =
A
1
∪ · · · ∪
A
n
is connected for
all
n
≥
1. Since the spaces
A
(
n
) have a point in common, namely any point of
A
1
, their union
A
(
n
) =
A
n
is connected by [1, Thm 23.3] again.
Ex. 23.3.
Let
A
∪
A
α
=
C
∪
D
be a separation. The connected space
A
is [Lemma 23.2] entirely
contained in
C
or
D
, let’s say that
A
⊂
C
.
Similarly, for each
α
, the connected [1, Thm 23.3]
space
A
∪
A
α
is contained entirely in
C
or
D
. Sine it does have something in common with
C
,
namely
A
, it is entirely contained in
C
. We conclude that
A
∪
A
α
=
C
and
D
=
∅
, contradicting
the assumption that
C
∪
D
is a separation
Ex. 23.4 (Morten Poulsen).
Suppose
∅
A
X
is open and closed. Since
A
is open it follows
that
X

A
is finite. Since
A
is closed it follows that
X

A
open, hence
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 Fall '08
 Brown
 Topology, Topological space, General topology, Ex., Ex 17.19

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