Ph.D. EXAM: TOPOLOGY June 1997

1. Prove that every path connected space is connected.

2. Let A and B be compact subsets of a Hausdorff space X. Prove that A ∩ B is compact. Hint: Do not use the definitions of “compact” and “Hausdorff.” Warning: The statement is not in general true for non-Hausdorff spaces.

3. Prove that if X is a metric space which has a countable dense subset, then X satisfies the second axiom of countability.

4. Let G be a group of self-homeomorphisms of a space X. Define an equivalence relation on X by setting x ∼ x′ if and only if there exists a g ∈ G such that g(x) = x′. Let Y be the set of equivalence classes, and let p : X → Y be the function which takes x to its equivalence class. Give Y the quotient topology induced by p.

(a) Prove that p is an open map.

(b) Prove that if G is finite, then p is a closed map.

Hint: For A ⊆ X one has p^{−1}(p(A)) = ⋃_{g∈G} g(A).

5. Prove that every retract A of a Hausdorff space X is closed in X. Hint: Show that X − A is open.

6. Let M_{g} denote the connected sum of g tori; let N_{g} denote the connected sum of g projective planes. Classify each of the following as an M_{g} or N_{g}, specifying g.

(a) The orientable surface (without boundary) having Euler characteristic −14.

(b) The connected sum of a torus and a Klein bottle.

7. Let X be a path connected space. Let x, y ∈ X. If γ is a path class from x to y, then denote by γ̂ : π_{1}(X, x) → π_{1}(X, y) the change of basepoint isomorphism given by γ̂(α) = γ^{−1} · α · γ. Prove that γ̂ = δ̂ for every pair of path classes γ and δ from x to y if and only if π_{1}(X, x) is abelian.

8. Let X ∨Y denote the space obtained by joining the spaces X and Y together at a single point. Write down the fundamental groups of the following spaces.

(a) (S^{1} ∨ S^{1}) × S^{1}

(b) S^{1} ∨ (S^{1} × S^{1})

9. Let p : X̃ → X be a covering map, where X̃ and X are path connected and locally path connected. Let x̃ ∈ X̃ and x = p(x̃). Prove that p : X̃ → X is one to one if and only if p_{∗} : π_{1}(X̃, x̃) → π_{1}(X, x) is onto.

10. Prove that R^{2} and R^{3} are not homeomorphic.