Calculus on manifolds

Okay, so I'm reading Spivak's "Calculus on manifolds" and in exercise 1-21b you have to prove that in an euclidean space if A is a closed set, B is compact and their intersection is empty, there is d>0 such that |y-x|>=d for every x in A and y in B. It also says this doesn't work if B is closed but not compact. Now, no matter how I approach this it seems to me that the sets only need to be closed. Anybody knows what I'm doing wrong?

Ok, A is closed, B is compact A n B is empty. in an euclidean space X

Consider the function d:B -> R (the reals)

d(b) = inf d(a,b) over a in A.

Now if d(b) = 0 then this implies there is a limit point of A in B, but since A is closed this implies their intersection is non-empty. Therefore for all b in b there exist a k_b s.t d(b) > k_b > 0.

Now we need to show inf k_b > 0.

So B compact, lets take a e-cover ( i.e a cover of the open e balls around each b in B). We now take the finite subcover let's call the point c_i for 0< i <=n

Now, notice that min d(c_i) exists and is bigger than 0.

Note also that d(c_i) > e for all c_i as otherwise A n B would not be empty.

Therefore min d(c_i) > e + k, some k

However note that d(b) > d(c_b) - e where c_b is the closest c_i to the point b, for all b in B.

Therefore d(b) > k , some k, for all b in B.


That was fun. I got about my only marks on last years exams on metric and topological spaces.