Combinatory Topology of Convex Regions.

5 Cf. abstract, Bull. Amer. Math. Soc., 34, 1928 (551). Whyburn's paper, "Concerning Points of Continuous Curves Defined by Certain im Kleinen Properties," which contains this example in full, is to appear in an early issue of Math. Ann. My ability to reproduce it here is due to Professor Whyburn's kindness in communicating it to me. 6 Hausdorff, Grundzuge der Mengenlehre, Leipzig, 1914, p. 213. 7 If K is a set of points, by K' we denote the set consisting of K together with its limit points. 8 These PROCZECDIGS, 12, 1926 (761-767). 9 Compare the first part of this proof with that given by Whyburn for the case cited above. 10 Wilder, R. L., Bull. Amer. Math. Soc., 32, 1926 (338-340). It is clear from its proof that this theorem is true in any topological space. 21 Moore, R. L., Math. Zs., 15, 1922 (254-260). 12 Wilder, R. L., Bull. Amer. Math. Soc., 34, 1928 (649-655). The theorem quoted here was proved for euclidean spaces, but it is clear that it holds in any locally compact metric space, and, indeed, may be proved directly by use of the imbedding theorem employed in proving Theorem 4 below, and hence so that N is of the same dimension (Menger-Urysohn) as H, etc. 13 Kline, J. R., abstract, BuU. Amer. Math. Soc., 34, 1928 (263). 14Wilder, R. L., these PROCXEDINGS, 11, 1925 (725-728). Although the proof is given in the terminology of the plane, it is clear that the theorem referred to is true in any localy compact metric space; in fact, the necessity proof holds in any topological space.