A Note on the Definition of Arc-sets

The interesting subsets of a Peano space which were later called the arc-sets or A -sets were defined independently and in a different manner by G. T. Whyburn and the author, and each studied the properties of these sets which have proved so useful in the later theory. The similarity of the properties led shortly to the conjecture that a relation existed between the two definitions, and investigation proved that they were equivalent. These are the properties (1) and (3) of the theorem of the present note. Two more equivalent properties, the properties (2) and (4) of the present note, were observed immediately. The present note introduces two new properties and shows that all six are equivalent, so that any one may be taken as the definition of the arc-sets or A -sets.