A Property Which Characterizes Continuous Curves.

Let M be a point set, C1 and C2 mutually exclusive closed subsets of M, and K a connected subset of M such that (1) K contains no points of the set C1 + C2 and (2) both C1 and C2 contain limit points of K. Then K, together with its limit points in C1 + C2, will be caUed a set K(C1,C2)M. If M has the property that for every two of its mutually exclusive closed subsets Cl, C2 and every connected set N containing points of. both C1 and C2, there exists some set K(C1,C2)M which has a point in common with N, then M will be called normally connected. That not every connected point set, nor, indeed, every continuum, is normally connected, is shown by the following example: Let M be the set of points on the curve