Concerning Arcwise Connectedness and the Existence of Simple Closed Curves in Plane Continua

Charles L. Hagopian
1970 Transactions of the American Mathematical Society  
Introduction. If a compact metric continuum is locally connected, then it is arcwise connected. However, a semi-locally-connected compact metric continuum, even when lying in the plane, may fail to be arcwise connected [8]. A continuum M is said to be aposyndetic at a point p of M with respect to a set Aî n M-{p} if there exist an open set U and a continuum 77 in M such that p e U^HcM-N. A continuum M is said to be aposyndetic at a point p if for each point q in M-{p}, M is aposyndetic at p
more » ... posyndetic at p with respect to q. If M is aposyndetic at each of its points, then M is said to be aposyndetic. A compact continuum is aposyndetic if and only if it is semi-locally-connected [2]. A plane continuum M is connected im kleinen at a point x of M if and only if for each pair of points y and z in M-{x}, M is aposyndetic at x with respect to {y, z} [A]. In this paper arcwise connectedness is established for certain nonconnected im kleinen aposyndetic compact plane continua. If an aposyndetic compact plane continuum A7 contains a finite set of points £such that for each point x in M-F, there exist two points y and z in £such that M is not aposyndetic at x with respect to {y, z}, then M is arcwise connected. It is also proved that each point of M is in a simple closed curve which is contained in M and if the set £ consists of two points, then M is cyclicly connected (that is, for any points a and b in M, there is a simple closed curve in M which contains {a, b}). Arcwise connectedness is also established for certain nonaposyndetic plane continua. If a compact plane continuum M is semi-locally-connected at all except a finite number of its points and is such that for each point x in M, M is either not aposyndetic or not semi-locallyconnected at x, then M is arcwise connected. For a point xofa continuum M, F. B. Jones defines Kx to be the closed (not necessarily connected) set consisting of x and all points y in M-{x} such that M is not aposyndetic at x with respect to v [3, Theorem 2]. Here it is proved that if M is a compact plane continuum such that for each point x of M, the set Kx is finite, and Af is either not semi-locallyconnected or not aposyndetic at x, then each point of M is in a simple closed curve which is contained in M. Let M be a compact plane continuum which contains a point y such that for each point x in M-{y}, M is semi-locally-connected at x and
doi:10.2307/1995202 fatcat:7ot6bph3pra75ezt3tvyplrene