### Certain Theorems Relating to Plane Connected Point Sets

Transactions of the American Mathematical Society, Vol. 24, No. 2 (Sept., 1922), 144-162

Various parts of this paper were presented to the Society on October 25, 1919, December 28, 1920, and February 26, 1921.

Introduction

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 M_{1} and M_{2} 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 the M(bar), the set of points common to M_{1} and M_{2}, be not connected.