Just wondering if someone could help me with this problem.
For two disjoint compact subsets of R^n, show that there is a minimum positive distance between the points of each set.
Here is an outline.
Suppose that the two subsets are A & B and that D(A,B)=0.
Use the definition of distance between sets to get a sequence of distinct points in A that must converge to a point in B. That happens due to compactness. That will give you a contradiction.
Here is an outline.
Suppose that the two subsets are A & B and that D(A,B)=0.
Use the definition of distance between sets to get a sequence of distinct points in A that must converge to a point in B. That happens due to compactness. That will give you a contradiction.
Thanks a lot for this. Does it make sense to also say that since A,B are compact, the function that describes a path between a point in A and a point in B must have a minimum and a maximum because it is a function on a compact set?