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Serge Ballif
MATH 527 Homework 3
September 21, 2007
Problem 1.
Let
X
be a compact Hausdorff space. Let
K
1
⊃
K
2
⊃
...
be a sequence of closed
connected subsets of
X
. Let
K
=
∩
∞
i
=1
K
i
. Show that
K
is connected.
We note ﬁrst that
X
is normal since it is compact Hausdorﬀ. Also,
K
and each
set
K
i
are closed as an intersection of closed sets. Furthermore,
K
is normal as a
closed subspace of a normal space. Seeking a contradiction we assume that there
is a separation of
K
consisting of closed sets
A
and
B
(
A
∪
B
=
K
and
A
∩
B
=
∅
).
By the Urysohn lemma there exists a continuous function
f
:
K
→
[0
,
1] such that
f
(
x
) = 0 for every
x
in
A
and
f
(
x
) = 1 for every
x
in
B
. By the Tietze extension
theorem the map
f
can be extended to a continuous map
¯
f
:
X
→
[0
,
1]. We
now consider the sets
¯
f
(
K
i
)
⊂
[0
,
1]. Each set
¯
f
(
K
i
) is connected as a continuous
image of a connected set. Furthermore, since each set
K
i
contains the sets
A
and
B
, we know that
¯
f
(
K
i
) contains the points
f
(
A
) = 0 and
f
(
B
) = 1. This means
that
¯
f
(
K
i
) = [0
,
1] for all
i
. Thus for each
i
there is an element
k
∈
K
i
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 Fall '07
 ROTMAN,REGINA
 Math, Topology, Sets

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