5310 PRELIM
Introduction to Geometry and Topology
August 2012
You may use any result that has been proven in class, unless the question
directly asks you to prove the result. Please, state the results you are using,
and check that the assumptions are satisfied.
1. Show that R2with the Euclidean metric topology coincides with the
product topology of R×R.
2. Suppose f:X→Yis a continuous bijection, Xis compact, and Yis
Hausdorff. Prove that fis a homeomorphism.
3. Let fbe a continuous map between topological spaces Xand Y. Prove
or give a counterexample: if Xis locally compact, that is, Xis a
Hausdorff space such that every point of Xhas an open neighbourhood
whose closure is compact, then f(X) is locally compact.
4. Determine the number of (not necessarily connected) 3 : 1-coverings
of S1×S1.
5. Let Cbe the union of two unlinked circles {(x, y, z):(x−2)2+y2=
1, z = 0}and {(x, y, z) : (x+ 2)2+y2= 1, z = 0}in R3. Show that
π1(R3\C)is a free group on two generators.
6. Suppose p:Y→Xis a covering such that p−1(x) is finite and
nonempty for all x∈X. Show that Yis compact Hausdorff if Xis
compact Hausdorff.
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