GEOMETRY/TOPOLOGY QUALIFYING EXAM
August 2008
INSTRUCTIONS:
•You must work on all problems below.
•Use a separate sheet of paper for each problem and write only on
one side of the paper.
•Write your name on the top right corner of each page.
Problem 1. Let R`be the real line Rwith the lower limit topology, gen-
erated by half-open intervals [x,y ) (also called the Sorgenfrey line). Recall
that a space is called Lindel¨of if every open cover has a countable subcover.
a. Prove that R`is first countable, Lindel¨of, and separable, but not second
countable.
b. Let R2
`be the plane R2with the product topology R`×R`. Show that
R2
`is first countable and separable, but not Lindel¨of.
Problem 2. In this problem all spaces are assumed to be T1.
a. Fix a subbasis Sof X. Prove that Xis completely regular (T31
2) if and
only if for each point x∈Xand neighborhood V∈Swith x∈V,
there exists a map f:X→[0,1] such that f(x) = 0 and f(y) = 1 for
y∈X−V.
b. Show that the arbitrary product of completely regular spaces is com-
pletely regular. (Hint: Use a.)
c. Show that any subspace of a completely regular space is completely reg-
ular.
Problem 3. Recall that g:X→Yis called a proper map if g−1(C) is
compact whenever C⊂Yis compact. Show that if a map f:X→Yis
closed and f−1(y) is compact for all y∈Y, then fis proper.
Problem 4. Atopological manifold Mis a Hausdorff and second countable
space which is locally homeomorphic to an open subset of an Euclidean
space. Show that a topological manifold is metrizable. Is Mparacompact?
Is Mnormal?
Problem 5. Given u∈Rnand c∈R, define
S:= {(x,y)∈Rn×Rm| hx,ui2=||y||2+c},
where hx,uiis the inner product of xand u, and ||y|| is the norm of y.
For which constants cis Sa smooth submanifold of Rn×Rm? Prove your
assertion.
Problem 6. Let Xand Ybe smooth manifolds.
a. Show that T(x,y)(X×Y) = TxX×TyY.
b. Define the tangent bundle T(X) of X. Describe how T(X) is given the
structure of a smooth manifold.
August 2008 Geometry/Topology Qualifying Exam Page 1 of 2
