Topology Qualifying Exam
August 23, 2010
Do all of the following problems.
1. Let
X
be a topological space.
(a) (5 points) Define what it means to say that
X
is
regular
.
(b) (10 points) Give a complete proof that a product of two regular spaces is
regular.
2. Let X be a space.
(a) (5 points) Define what it means to say that
X
is
normal
.
It is easy to show that a normal space is regular. On the other hand the space
R
2
l
, the socalled
Sorgenfrey plane
, is a space that is regular, but not normal.
Here
R
l
is the real line with the lower limit topology.
(b) (5 points) Prove that
R
2
l
is regular.
(c) (10 points) Prove that
R
2
l
is not normal.
3. Let
X
be a topological space.
(a) (5 points) Define what it means to say
X
is
metrizable
.
(b) (10 points) Let
R
be the real line with the usual topology.
Let
R
ω
be the
product of countably many copies
R
and give
R
ω
the product topology. Show
that
R
ω
is metrizable.
4. Let
X
be a topological space.
(a) (10 points) Give two necessary conditions, other than separation properties,
for
X
to be metrizable.
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 Spring '11
 NA
 Logic, Topology, Ri, Topological space, Hausdorff space, Rω

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