Middle East Technical University Spring 1998 Math 422 Elementary Topology and Geometry

MIDTERM II

MAY 12, TIME: 90 MIN

Problem 1. (7 pts) Prove that if f: A → ℝ^{m} is a continuous map and A ⊂ ℝ^{n} is path connected. then so is f(A).

Problem 2. (5 pts) Determine the connected components of A = (ℚ × ℝ) ∪ (ℝ × {0}) ⊂ ℝ^{2}.

Problem 3. (10 pts) Given two disjoint compact sets, A, B ⊂ ℝ^{n}, prove that there exists ϵ > 0 such that |x − y| > ϵ for any points x ∈ A and y ∈ B.

Problem 4. (10 pts) Represent X as a quotient of Y in the following cases (1) X = S^{1}, Y = [0, 1]

(2) X is a torus, Y is a 2-disc

Problem 5. (15 pts) Which ones of the following sets are – surfaces

– compact surfaces with boundary

– non-orientable surfaces

(1) S^{1} × [0, 1),

(2) S^{1} × S^{2},

(3) [0, 1] × [0, 1],

(4) D^{2} × S^{2},

(5) T^{2}#T^{2},

(6) M#T^{2} (M is a Möbius band),

(7) ℝP^{2}#T^{2},

(8) a Klein Bottle with a hole

(9) a punctured 2-sphere

(10) {(x, y, z) ∈ ℝ^{3} | y^{2} = x^{2} + z^{2}}

(11) {(x, y, z) ∈ ℝ^{3} | x = y^{2} − z^{3}}

Problem 6. (20 pts) Assume that D_{1}, D_{2} ⊂ ℝ^{2} are topological 2-discs and h: ℝ^{2} → ℝ^{2} is a homeomorphism such that h(∂D_{1}) = (∂D_{2}). Prove that h(D_{1}) = D_{2}.

Problem 7. (20 pts) Give a gluing scheme that yields (1) an orientable surface of genus 2 with a hole,

(2) T^{2}#ℝP^{2}

(3) ℝP^{2} with 2 holes.

Problem 8. (20 pts) For the surfaces F sketched below (1) determine the Euler characteristic;

(2) point out which surfaces are homeomorphic, if there are any;

(3) determine the place of the surfaces in the classification list if they are closed (F_{g}, N_{k}) and point out the number of boundary components if they have any.

Explain your answer.