A converse of the Jordan-Brouwer separation theorem in three dimensions
R L Wilder · Transactions of the American Mathematical Society · 1930
A n example of a simply connected surface bounding a region which is net simply connected, Proceedings of the National Academy of Sciences, vol. 10 (1924), pp.8-10.f It will be shown below, in the Appendix, that uniformly homologous to zero as a condition relative to the 0-chains of a domain is equivalent to the condition uniform connectedness im kleinen for the domain.Í The earliest characterization of which the author knows was announced by R. L. Moore and J. R. Kline, their abstract being in the Bulletin of the American Mathematical Society, vol.28 (1922), p. 380.The same definition is given by Miss I. Gawehn (except for the greater generality of the space considered), in Über unberandete 2-dimensionale Mannigfaltigkeiten, Mathematische Annalen, vol.98 (1927-28), pp.321-354.Cf. also C. Kuratowski, Une caractérisation topologique de la surface de la sphère, Fundamenta Mathematicae, vol.13 ( 1929), pp.307-318, as well as abstracts by L. Zippin, Bulletin of the American Mathematical Society, vol.35 (1929), p. 154 and p. 293.* R. L. Moore, Mathematische Zeitschrift, loe.cit.t The word "chain" is here used in the ordinary point-set theoretic sense; i.e., if e is a positive number, and P and Q are points, an «-chain from P to Q is a set of points Pi, Pi, • • • , P", where Pi=P and P"=e, and p(Pi, Pi+l)