Download Math 571 Qualifying Exam 1. Let (Y,T ) be a topological space, and

Math 571
Qualifying Exam
1. Let (Y, T ) be a topological space, and let X be a set such that there exists a surjective function
f : X −→ Y.
Let T1 be the smallest topology in X that makes f a quotient map; T2 the smallest topology in X
that makes f continuous; T3 the smallest topology in X that makes f an open map; T4 the smallest
topology in X that makes f an closed map; T5 the smallest topology in X that makes f an open and
closed map; and T6 the smallest topology in X that makes f a closed and continuous map.
Compare four pairs of topologies from among {T1 , T2 , T3 , T4 , T5 , T6 } (your choices), and justify your
answers by examples if needed.
2.
a) Show that a topological space A is locally connected if and only if each connected component of
A is open in A.
b) Let X and Y be topological spaces such that X is locally connected. Let f : X −→ Y be a
continuous, surjective, closed map (taking closed sets to closed sets). Show that Y is locally
conected.
3. Let X be a compact Hausdorff space, and let A and B be disjoint closed nonempty sets of X. Show
that there exist disjoint open sets K and W such that A ⊂ K and B ⊂ W .
4. Let C be the boundary of the unit cube [0, 1] × [0, 1] × [0, 1] (i.e., the union of the six faces). Label
a, b, c, d the four edges of the top face, oriented clockwise. Identify a with −c, and b with −d, and call
e the resulting quotient space.
C
e
Use the van Kampen Theorem to compute the fundamental group of C.
5. Suppose f : P2 × P2 −→ T2 is continuous (for P2 the projective plane, and T2 the torus).
a) Show that f∗ : π1 (P2 × P2 ) −→ π1 (T2 ) is the trivial homomorphism.
b) Show that f is null-homotopic; that is, it is homotopic to a constant map (you may use the general
lifting property).
6. Let E and Y be Hausdorff, connected and locally path connected, so that E is compact and simply
connected. Suppose there is a covering map p : E −→ Y . Show that for any base point b ∈ Y , we have
that π1 (Y, b) is a finite group.
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