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?
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Anonymous2008-08-06 11:43
I'd imagine the reason is that you could have some sort of intersection at infinity that doesn't really count so to speak.
I can't imagine this is a counterexample, but it'll be similar,
Take the closed sets A = (x,y) s.t y>=1/x x>0
B = (x,y) s.t y>-1/x x<0
Then there's no minimal distance.
Write out your proof with them just being closed, I imagine it'd be easier for me to see the flaw there then to write out a counterexample. I'll sketch a quick proof of my own if I have time.
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.
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Anonymous2008-08-06 14:14
>>4
Thanks. Wow, that was pretty cool. Topology proofs always give me this " OMG AWESOME" feeling.
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Anonymous2008-08-06 14:40
I don't understand it.
I need more Intelligence.
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Anonymous2008-08-06 16:42
ITT high school kid thinks he is a genius and can unravel the mysteries of the universe by asking questions, on 4chan, to which if he got the correct answer he would not understand it until his 2nd year of graduate school.
many people reply
I answered his question, and I only used methods I learnt in my first year of uni.
Frankly I'm annoyed I couldn't work out how to prove my function d was continuous, as obviously that would have completed the proof in a different way. just had a mental block.