the minimum distance between a compact and a connected set is achieved by some pair of points in the sets
I don’t think this is true. For a counterexample in the plane, let A be the set consisting of the point (2,0) and let B be the open unit disk centered at the origin. A is compact, B is connected, and the infimum of {distance from x to y where x is in A and y is in B} is 1. But the distance from A to any point in B is strictly greater than 1.
I don’t think this is true. For a counterexample in the plane, let A be the set consisting of the point (2,0) and let B be the open unit disk centered at the origin. A is compact, B is connected, and the infimum of {distance from x to y where x is in A and y is in B} is 1. But the distance from A to any point in B is strictly greater than 1.
Thanks, it looks like I accidentally typed “connected” instead of “closed”; fixed.