Math 468

Problem set 2, due Friday, January 30

1. Prove that in Rⁿ, the only sets which are both open and closed are the empty set and all of Rⁿ. (If you can't figure this out in general, try to do it when n=1.)

2. If U₁, U₂, ..., U_n are all open, is their intersection? Prove or give a counterexample.

3. If U₁, U₂, ..., U_n, ... are all open, is their intersection? (The difference between this problem and the previous one is that in this case there are infinitely many sets U_i; in problem 2 there were only finitely many.)

4. Suppose f:Rⁿ -> R^k is a function. What does it mean for f to be continuous? Try to explain it in words; also try to give a precise mathematical definition.

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