DOCTORAL EXAM (OLD OPTION) - TOPOLOGY - JANUARY 1998

1. Prove that X is Hausdorff if and only if the diagonal Δ = {(x, x)|x ∈ X} is closed in X × X.

2. Let {C_{n}} be a sequence of closed, non-empty subsets of a compact space X such that C_{n} ⊇ C_{n+1} for all n. Prove from the definition of compactness that ∩C_{n} ≠ Ø.

3. Let U be a connected, non-empty open subset of R^{2}. Prove that U is path-connected.

4. Prove that every separable metric space satisfies the second axiom of countability.

5. Give examples of the following. Support your answers.

(a) A space with trivial fundamental group.

(b) A space with a finite non-trivial fundamental group.

(c) A space with an infinite cyclic fundamental group.

(d) A space with a non-abelian fundamental group.

6. For i = 1, 2, let F_{i} be the free group on the set S_{i} with respect to the function φ_{i} : S_{i} → F_{i}. Assume that S_{1} and S_{2} are disjoint. Let S = S_{1} ∪ S_{2}. Let F be the free product of F_{1} and F_{2} with respect to the homomorphisms ψ_{i} : F_{i} → F. Define φ : S → F by φ|_{S_{i}} = ψ_{i} ∘ φ_{i}. Prove that F is the free group on S with respect to φ.

7. Let T be a triangulation of S^{2} such that each vertex has valence either 5 or 6. (Recall that the valence is the number of edges meeting the vertex.) Prove that T has exactly 12 vertices of valence 5.

8. Let T be the torus S^{1} × S^{1}.

(a) Give an example of a covering map p : X → T such that π_{1}(X) is isomorphic to Z.

(b) Give an example of a 3-sheeted covering map p : Y → T with Y connected.

(c) Up to isomorphism how many 3-sheeted covering maps p : Y → T with Y connected are there? Justify your answer.