1st December 2004
Munkres
§
32
Ex. 32.1.
Let
Y
be a closed subspace of the normal space
X
. Then
Y
is Hausdorﬀ [Thm 17.11].
Let
A
and
B
be disjoint closed subspaces of
Y
. Since
A
and
B
are closed also in
X
, they can
be separated in
X
by disjoint open sets
U
and
V
. Then
Y
∩
U
and
V
∩
Y
are open sets in
Y
separating
A
and
B
.
Ex. 32.3.
Look at [Thm 29.2] and [Lemma 31.1]. By [Ex 33.7], locally compact Hausdorﬀ spaces
are even completely regular.
Ex. 32.4.
Let
A
and
B
be disjoint closed subsets of a regular Lindel¨
of space. We proceed as
in the proof of [Thm 32.1]. Each point
a
∈
A
has an open neighborhood
U
a
with closure
U
a
disjoint from
B
. Applying the Lindel¨
of property to the open covering
{
U
a
}
a
∈
A
∪ {
X

A
}
we
get a countable open covering
{
U
i
}
i
∈
Z
+
of
A
such that the closure of each
U
i
is disjoint from
B
.
Similarly, there is a countable open covering
{
V
i
}
i
∈
Z
+
of
B
such that the closure of each
V
i
is
disjoint from
A
. Now the open set
S
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 Fall '08
 Brown
 Topology, Sets, General topology, disjoint open sets

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