In a metric space , let and be disjoint closed subsets of . Prove that disjoint open sets and of such that and .
This is not true: Take the graph of in it is like two hyperbolas that get arbitrarily close as you approach (they both go to infinity). It is clearly closed (each 'hyperbola' is also closed), and the two 'hyperbolas' are disjoint but . for this to work either or must be compact
That is correct: if then . But putnam120's argument doesn't work for the reason pointed out by Jose27. The distance from B to A can be zero, as shown by the example of the two components of the graph of y=1/x^2. But the distance from any individual point in B to A must be strictly positive.
My argument doesn't work because I took the over all pairs and . But Opalg's argument you look at a particular . To see that if just use the limit point definition of a closed set.
I have a question for Jose27: How are you able to see that my original argument would work if one of A or B were compact?