Ph.D. COMPREHENSIVE EXAM - TOPOLOGY - JUNE 1995

1. (12 points) Let X be a Hausdorff space. Suppose A ⊆ X is a retract of X. Prove that A is closed in X.

2. (12 points) Suppose f : X → Y is continuous and onto, X is compact, and Y is Hausdorff. Prove that f is a quotient map.

3. (12 points) Prove that every metrizable Lindelöf space X is second countable. Hint: Let U_{n} be a covering of X by open 1/n-balls.

4. (12 points)

(a) State the Urysohn Lemma.

(b) Prove that every connected normal space having at least two points is uncountable.

5. (12 points) 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 surfaces as an M_{g} or N_{g}, specifying g.

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

(b) The compact surface S with χ(S) = −7.

(c) The compact, orientable surface S with χ(S) = −8.

6. (12 points) Let T be a torus and P a projective plane. Give presentations for the fundamental groups of the following spaces. (Don’t just say what the groups are algebraically; give presentations for them.)

(a) T × P

(b) T #P, where “#” denotes connected sum.

(c) T ∨ P, where “∨” denotes the gluing of the two spaces together at a single point.

7. (12 points) Define p : R^{3} → R^{2} × {0} by p(x, y, z) = (x, y, 0), i.e. p is just projection onto the x-y plane. Let D = {(x, y, 0) | x^{2} + y^{2} ≤ 1} be the unit disk in the x-y plane. Suppose E is a disk embedded in R^{3} (perhaps in a very peculiar manner) such that ∂E = ∂D = {(x, y, 0) | x^{2} + y^{2} = 1}, i.e. the unit circle in the x-y plane. Prove that (0, 0, 0) ∈ p(E). Hint: Consider R^{3} minus the z axis. Remark: With slightly more work one can show that D ⊆ p(E).

8. (12 points) Let p : X̃ → X be a covering space. X̃ and X are assumed to be path connected and locally path connected. Let x, y ∈ X. Prove that there is a bijection φ : p^{−1}(x) → p^{−1}(y).

9. (4 points) State the Seifert-Van Kampen Theorem for the case of a space X which is the union of two subspaces U and V. Be sure to state all the hypotheses.