connected plane continua

A continuum M is said to be X connected if any two distinct points of M can be joined by a hereditarily decomposable continuum in M. Recently this generalization of arcwise connectivity has been related to fixed point problems in the plane. In particular, it is known that every A connected nonseparating plane continuum has the fixed point property. The importance of arcwise connectivity is, to a considerable extent, due to the fact that it is a continuous invariant. To show that A connectivity has a similar feature is the primary purpose of this paper. Here it is proved that if Ai is a A connected continuum and / is a continuous function of M into the plane, then f(M) is A connected. It is also proved that every semiaposyndetic plane continuum is A connected. Introduction. A nondegenerate metric space that is both compact and connected is called a continuum. It is known that every plane continuum that has a hereditarily decomposable boundary and does not separate the plane has the fixed point property [l]. Recently the author [4] proved that every arcwise connected nonseparating plane continuum has a hereditarily decomposable boundary. Hence all arcwise connected nonseparating plane continua have the fixed point property. In [7] it is pointed out that the author's theorem remains true if the word "arcwise" is replaced by "A". In fact, in [7] it is proved that a plane continuum that does not have infinitely many complementary domains is A connected if and only if its boundary does not contain an indecomposable continuum. This paper is primarily concerned with the following questions: (1) What other theorems about arcwise connected continua also hold for A connected continua? (2) Are there general properties, other than arcwise connectivity for plane continua, that imply A connectivity? Received by the editors September 13, 1971 and, in revised form, September 6, 1973. AMS (MOS) subject classifications (1970). Primary 54C05, 54F25, 54F60, 57A05; Secondary 54F15.