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In their paper On the most general plane closed point set through which it is possible to pass a simple continuous arc, R. L. Moore and J. R. Klinet prove that it is possible to pass a simple continuous arc through every closed and bounded set M having the property that every closed and connected subset of M is either a single point or an arc t such that no point of t, with the exception of its end points, is a limit point of M t. It is clear, however, that in order that a simple continuous arc may be drawn in such a way as to contain at least one point of every maximal connected subset of a point set M, it is not necessary that the set M be of the particular type satisfying their theorem. In this paper I shall make a study of certain conditions which a point set must satisfy in order that a simple continuous arc or an open curve may be drawn in such a way that the set in question plus that arc or curve will be connected.

Concerning point sets which can be made connected by the addition of a simple continuous arc