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Homework 9

Math 447:  Real Variables

Exercise 1 Let X be a metric space. Prove that X is connected if and only if the only subsets that are both open and closed are ∅ and X .

Note: do not assume path connectivity.

Exercise 2 Let n > 0. The n-sphere is the subset Sn ⊂ Rn+1  deined as

Sn = { x 2 Rn+1 | d2 (x, 0) = 1}.

Assume f : Sn → R is a continuous function. Prove that there exists x 2 Sn  such that f (x) = f (-x).  Conclude that f cannot be injective.

Exercise 3 Let X and Y be connected metric spaces. Prove that X × Y is connected. Note: do not assume path connectivity.

Exercise 4 Let O be a connected open set in Rn , n > 1.  Suppose f : O → R is continuous.  Prove that f cannot be injective.