TOPOLOGY QUALIFYING EXAM
AUGUST 2009
There are six problems. Begin your answer to any problem on a new page in your blue
book(s).
Make a space between answers to separate parts
ofa question tofacilitate
grading. All answers must be justified with proofs.
The relative point values are
indicated.
1 (12 pts) Let
Y
be an ordered set in the order topo10gy. Let
f, g
:
X
~
Y
be
continuous.
a) Show that the set
{x
I
f(x);:::: g(x)}
is closed in
X.
b) Let
h
:
X
~
Y
be the function
hex)
=
max{f(x), g(x)}.
Show that
h
is continuous.
2 (18 pts) a) Show
if
Y
is Hausdorff, then
y
X
(=
the space
of all functions from
X
to
Y)
with the compact open topology is also Hausdorff.
b) Let /" :
R
~
R (R
=
the real numbers with the usual topology) be defined by
1
I
2
2
yn
x
if[xl
<
n
n
o
iflxl;::::n
Show that
{fJdoes
not converge to the constant function
g(x)
=
1 in the uniform
topology, but that it does converge to this constant function in the compact convergence
topology.
c) Let
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 Spring '11
 NA
 Topology, Hausdorff, compact open topology

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