15 January 2010 Geometry & Topology Qualifying Examination
Notation. Let Rndenote real n-space. Let Sn={x∈Rn+1 :|x|= 1}denote the
n-sphere in Rn+1.
Employ the summation convention: any repeated index appearing as a subscript
and superscript is summed over.
1.) Prove the Tube Lemma: Let Xbe a topological space and let Ybe a compact
topological space. Prove that if Wis an open subset of X×Y(in the product
topology) containing the “slice ”{x0} × Yfor some x0∈X, then there exists
an open neighborhood Uof x0in Xsuch that U×Y⊂W.
2.) (a) Prove that an arcwise connected topological space is connected.
(b) Let Xbe a normal, connected topological space containing more than
one point. Prove that Xis uncountable. (Hint : Use Urysohn’s lemma).
3.) Let R`denote the real line Rwith the lower limit topology (so-called Sorgenfrey
line), i.e. the topology consisting of all left-closed, right-open intervals [a, b).
Let R2
`denote the plane R2with the product topology R`×R`.
(a) Find the closure of the set (a, b) in R`.
(b) Prove that R`is not a locally compact space.
(c) Take the ‘anti-diagonal’ L={(x, −x) : x∈R`}of R2
`. Describe the
subspace topology that Linherits from R2
`.
(d) Prove that the space R2
`is not Lindel¨of (recall that a topological space X
is called Lindel¨of if every open cover of Xhas a countable subcover).
4.) Let R/Zbe the quotient space obtained from Rby the identification of the
subspace Zto a point. (Do not confuse this with the group quotient of Rby
Z.) Prove that this quotient space is not first countable.
5.) (a) Prove that for any locally finite family of sets {Vα}α∈Ain a topological
space Xone has
[
α∈A
Vα=[
α∈A
Vα.
(b) Recall that a topological space Xis called countably compact if every
countable cover of Xhas a finite subcover. Prove that a countably compact
paracompact space is compact.
6.) (a) State the definition of a smooth n–dimensional manifold.
(b) Set A=Rn+1\{0}. Given x, y ∈A, the condition
x∼yif and only if there exists λ6= 0 such that y=λx
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