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Topology Qualifying Exam
August 20, 2007
Do all of the following problems.
1. (20 points) Let
X
be a set.
(a) Deﬁne what it means to say that the collection
B
of subsets of
X
is a
basis
for
a topology on
X
.
(b) Suppose now that
X
is endowed with a topology
T
. Deﬁne what it means to
say that the subset
C
of
X
is
compact
(c) Let
R
be the real line. Give a basis for the lower limit topology on
R
. Let
R
l
denote
R
with this topology.
(d) Show that if
C
⊂
R
l
is compact, then
C
is closed and bounded.
(e) Show that the converse of (d) is false.
2. (20 points) Let
X
be a topological space with topology
T
.
(a) Deﬁne what it means to say that
X
is
connected
.
(b) Let
D
be a subset of
X
. Deﬁne what it means to say
D
is
dense
in
X
.
(c) Suppose
X
is connected and that
D
⊂
X
is dense. Let
T
′
be the topology on
X
with subbasis
T ∪ {
D
}
. Show that (
X,
T
′
) is connected.
3. (20 points) Recall that a topological space
X
is said to be
completely regular
if
onepoint sets are closed in
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 Spring '11
 NA
 Topology, Sets

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