Math 426, Winter 2011, Term 1
Final Exam
Tuesday, December 6, 2011
No books, notes or calculators
Problem 1.
Define the following terms:
(a) discrete topological space
(b) totally disconnected topological space
(c) regular topological space
(d) groupoid
(e) covering map
Problem 2.
Carefully state the following results:
(a) the tube lemma,
(b) the extreme value theorem,
(c) the Van Kampen theorem,
(d) the classification of covering spaces of a path connected space X in terms of
the fundamental group of X.
Problem 3.
Give examples of the following phenomena (without proofs):
(a) a quotient map which is not closed,
(b) a topological space not satisfying the second countability axiom
(c) a compact space which is not Hausdorff
(d) a connected subset of a topological space whose interiour is not connected.
(e) a locally path connected space, which is not semi-locally 1-connected,
(f) a covering map between path connected spaces, which is not regular (Galois),
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Math 426
Dec. 6, 2011
Problem 4.
Let Y be a topological space and let X be a locally compact Hausdorff space. Let
C(X, Y ) be the set of continuous maps endowed with the conpact-open topology.
Prove that the evaluation map
ev : X × C(X, Y ) −→ Y
(x, f ) 7−→ f (x)
is continuous.
Problem 5.
Decide if the following statements about two subsets A, B of a topological space X
are true or false. In each case, provide a proof or a counterexample.
(a) A ∪ B = A ∪ B.
(b) A ∩ B = A ∩ B.
Problem 6.
Define a topology on (0, 1) be taking open sets to be (0, 1), as well as all sets not
containing 0. It this topological space
(a) connected?
(b) path connected?
(c) compact?
(d) Hausdorff?
(e) second countable?
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Math 426
Dec. 6, 2011
Problem 7.
Suppose that f : X → S 2 is a covering map, and that X is connected. Prove that
f is a homeomorphism.
Problem 8.
Let X and Y be topological spaces, and f, g : X → Y two continuous maps. Further,
let θ : f ⇒ g be a homotopy from f to g.
f
X
θ
)
f∗
5
Y
Π1 (X)
g
θ∗
g∗
3
+
Π1 (Y )
Construct a natural transformation θ∗ : f∗ ⇒ g∗ , where f∗ , g∗ : Π1 (X) → Π1 (Y ) are
the groupoid morphisms induced by f and g on fundamental groupoids.
Problem 9.
(a) Prove that R and R2 are not homeomorphic.
(b) Prove that R2 and R3 are not homeomorphic.
(Hint: using connectedness and simply connectedness works.)
Problem 10.
Three surfaces are glued from the following three hexagons by pairwise identifying
edges according to the prescriptions in the following sketch:
Find the fundamental group of each surface. Decide which of the surfaces are
homeomorphic to each other.
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