6 Problems on Introduction to Theory of Functions II - Final Exam | MATH 766, Exams of Mathematics

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DEPARTMENT OF MATHEMATICS

UNIVERSITY OF KANSAS

Final MATH 766 - Spring 2011

Your Name:

Total (500)

(1) (85 points) Let {Uα}α be an open cover of a compact set K in a metric space (X, ρ). Show that there is  > 0, so that if p, q ∈ K : ρ(p, q) < , then there exists α = α(p, q, ), so that p, q ∈ Uα.

(3) (85 points) Consider the set of points S given by y^2 z = x^2. Determine for which points P on S, there is a neighborhood Vp of P and an open set D ⊂ R^2 , so that Vp ∩ S is diffeomorphic to D?

(4) (85 points) Let f, g : R^1 → R^1 be continuously differentiable. Suppose that f (0) = g(0) = 0, f ′(0) 6 = 0. Show that there exists an open subset V ⊂ R^2 containing (0, 0) and a continuously differentiable mapping ψ on V , so that ψ(u, v) = 0 if and only if (u, v) = f (x), g(x)) for some x near 0. Hint: Consider H : R^2 → R^2 , H(x, y) = (f (x), g(x) + y).

(6) (85 points)

(a) Let E and F be a nonempty closed subset and a nonempty compact subset of a metric space (X, d), with E ∩ F = ∅. Show that d(E, F ) = inf{d(e, f ) : e ∈ E, f ∈ F } > 0 (b) Give an example, where the conclusion (a) may fail, if F is closed, but not compact.