Proving Disjoint Closed Sets in Metric Space: A Compact Set

[giantsfan000]

Homework Statement

Suppose A and B are disjoint closed sets in the metric space X and assume
in addition that A is compact. Prove there exists ∆ > 0 such that for all
a ∈ A, b ∈ B, d(a, b) ≥ ∆

2. The attempt at a solution

I really don't have an attempt at a solution because I am 100% completely stuck. I just don't know where to start. I'm not looking for any answers or anything, if someone could just tell me the first step or so, or point me in the right direction, I'd be much obliged.

Thanks!

 

[Office_Shredder]

A proof by contradiction is a good place to look. What if d(a,b) can be arbitrarily small?

 

[giantsfan000]

I appreciated your prompt response to my question about topology. Thank you very much for that! Proof by contradiction does make sense. I hope you don't mind me asking a follow up question. Would the contradiction be that there is not an delta > 0 st d(a,b) > delta or there exists a delta > 0 st d(a,b) < delta. I really appreciate the help. This problem was assigned as a review, and I keep trying to do it, but it's really kicking my butt and I just keep getting stuck over and over again! I guess I just need to figure out my proper proof by contradiction statement and then see if I can somehow figure it out from there.

 

[Office_Shredder]

If there exists no delta as described in the problem, then for any value delta there exists a and b such that d(a,b)<delta

 

[giantsfan000]

Thank you so much for your help office shredder. I really do hate to ask a follow up again because I know you don't want to do too much of the problem for me, but I just feel so stupid when I try to do this problem. So I've got my statement set up that I want to contradict, but how exactly do I go about contradicting this? I know I need to make use of the fact that one of the sets is compact because if they were only both closed then I don't think this statement holds. So I want to use that they're disjoint, ones compact, ones closed, but how exactly would I do this? And again, I'm sorry to be asking so much, I usually try to just think about it until I figure stuff out, but this problem has just been kicking my butt all day long :(

 

[Office_Shredder]

Use the fact that you can find points in A and B that are arbitrarily close to construct a sequence of points in A and a sequence of points in B that do something special.

A common thing to do in situations like this is to consider finding points a and b for each choice of n with delta=1/n