Middle East Technical University Spring 1999 Math 422 Elementary Geometric Topology

MIDTERM I

MARCH 20, TIME: 90 MIN

Problem 1. (20 pts) Formulate definitions of (1) a boundary point of a set A ⊂ ℝ^{n},

(2) continuity of a map f: X → Y at a point x ∈ X,

(3) a compact set,

(4) a topological property,

(5) a connected component,

(6) a surface.

Problem 2. (8 pts) Prove that a square and a line segment are not homeomorphic.

Problem 3. (15 pts) Prove that Int(A∩B) = (Int A)∩(Int B) for any A, B ⊂ ℝ^{n}.

Problem 4. (10 pts) Prove that X is disconnected if and only if there exists a continuous surjection f: X → S^{0}. (S^{0} ⊂ ℝ is a null-dimensional sphere)

Problem 5. (10 pts) Prove that the sets ℤ ⊂ ℝ and ℚ ⊂ ℝ are not homeomorphic.

Problem 6. (10 pts) Consider a subset B ⊂ A ⊂ ℝ^{n} with the inclusion map f: B → A. Prove that f is an open map if and only if B is relatively open in A.

Problem 7. (15 pts) Sketch (or describe) an identification space Y

(1) X = [0, 1] × [0, 1] with the partition containing sets {(0, x), (x, 0)}, where x ∈ [0, 1], and one-point-sets {(x, y)}, x, y ∈ (0, 1].

(2) Y = ℝ^{2} ∪_{f} S^{2}, where f is a homeomorphism between a unit circle in ℝ^{2} and a big circle on a sphere S^{2}.

(3) Y = [−2, 3] ∪_{f} [−3, 2], where f: [0, 2] → [−2, 0], f(x) = −x.

Problem 8. (20 pts) Describe a partition on X with the identification space Y (1) X = D^{2} (a disc), Y = S^{2}

(2) X = S^{2}, Y = D^{2}

(3) X = [0, 1] ∪ [2, 3], Y ⊂ ℝ^{2} is a figure having the shape of letter “Y”.

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