G. T. Whyburn gave an
elementary inductive proof of the n-arc theorem for Peano spaces, which had
originally been proved by G. Nobeling and K. Menger. In the course of doing this he
gave a necessary and sufficient condition for there to be n disjoint arcs joining two
disjoint closed sets A and B in a Peano space S. In this paper we split the set A into
n disjoint closed subsets A1,A2,⋯,An and give a necessary and sufficient condition
for there to be n disjoint arcs joining A1∪ A2∪⋯An and B in S, exactly one arc
meeting each Ai. Our proof uses the inductive technique that Whyburn
introduced.
