Publisher
Springer Science and Business Media LLC
Reference47 articles.
1. The pointP of a connected setM is said to be a cut point ofM providedM−P is not connected. The notion of an im kleinen cut point of a continuum is contained implicitly in the works of P. Urysohn and R. L. Moore, and is closely approximated in that of R. G. Lubben and C. Zarankiewicz. Cf. P. Urysohn, Über im kleinen zusammenhängende Kontinua, Math. Annalen98 (1927), S. 296–308; [Urysohn uses the terms “unvermeidbar” (unavoidable) and “vermeidbar” (avoidable) to designate im kleinen cut points and non-im kleinen cut points, respectively]; R. L. Moore, Concerning Triods in the Plane and the Junction Points of Plane Continua, Proc. Ntl. Acad. of Sci.14 (1928), pp. 85–88; R. G. Lubben, Concerning Connectedness near a Point Set; and C. Zarankiewicz, Sur les points de division dans les ensembles connexes, Fund. Math.9 (1927), see proof of Theorem 14.
2. Cf. K. Menger, Grundzüge einer Theorie der Kurven, Math. Annalen95 (1925), S. 272–306.
3. Cf. K. Menger,loc cit., and P. Urysohn, Comptes Rendus175 (1922), p. 481. Urysohn uses the term ‘index of a point’ instead of the therm ‘order of a point’.
4. That this definition is equivalent for the case of continuous curves to the Wilder definition [R. L. Wilder, Concerning Continuous Curves, Fund. Math.7 (1925), pp. 340–377] was shown by H. M. Gehman. See Concerning End Point of Continuous Curves and other Continua, Trans. Amer. Math. Soc.30 (1928). In my thesis I showed that this definition is (for plane continuous curvesM) equivalent to the following simple one:P is an endpoint ofM provided thatP is an interior point of no arc inM.Cf. Concerning Continua in the Plane, Trans. Amer. Math. Soc.29 (1927), pp. 369–400, Theorem 12. For the extension of this and other results frequently used later ton-space see W. L. Ayres, Concerning Continuous Curves in a Space ofn Dimensions, Amer. Journal of Math.
5. Cf. W. Sierpinski, Comptes Rendus160, p. 305. Sierpinski defines a ramification point ofM as a pointP such thatM contains 3 continuaK, L andN, such thatK·L=K·N=L·N=P. It follows from a result of Menger's (Fund. Math.10) that the definition here given and Sierpinski's definition are equivalent for Menger regular curves. Rutt [Bull. Amer. Math. Soc.33 (1927), p. 411 (abstract)] has shown them equivalent for all plane continuous curves. It appears likely that they are equivalent for continuous curves inn-space.
Cited by
10 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献