Various parts of this paper were presented to the Society on October 25, 1919, December 28, 1920, and February 26, 1921.
A point set M is said to be connected if it cannot be expressed as the sum of two mutually exclusive point sets neither of which contains a limit point of the other. Sierpinski has shown that a closed, bounded, connected set of points in space of n dimensions cannot be separated into a countably infinity of closed point sets such that no two of them have a point in common. It will be shown in the present paper that for the case where n = 2, this theorem does not remain true if the stipulation that M is closed be removed. It will however be shown that a plane point set, regardless of whether it be closed or bounded, which separates its plane cannot be expressed as the sum of a countably infinity of closed, mutually exclusive point sets, no one of which separates the plane. Of the other results established, the principal one is that if M1 and M2 are closed, connected, bounded points sets, neither of which disconnects a plane S, a necessary and sufficient condition that their sum, M, shall disconnect S is that M(bar), the set of points common to M1 and M2, be not connected.