Vol. 31, No. 1, 1969

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Concerning continua not separated by any nonaposyndetic subcontinuum

Eldon Jon Vought

Vol. 31 (1969), No. 1, 257–262

Certain theorems that apply to compact, metric continua that are separated by none of their subcontinua can be generalized and strengthened in those continua that are separated by none of their nonaposyndetic subcontinua. For those of the former type, if the continuum is aposyndetic at a point, it is locally connected at the point. The same conclusion is possible if the continuum is not separated by any nonaposyndetic subcontinuum. Also, if a continuum is separated by no subcontinuum and cut by no point, it is a simple closed curve. A second result of this paper is to prove that if no nonaposyndetic subcontinuum separates and no point cuts the continuum, then it is a cyclically connected continuous curve; in fact this yields a characterization of hereditarily locally connected, cyclically connected continua.

A third theorem characterizes an hereditarily locally connected continuum as an aposyndetic continuum that is separated by no nonaposyndetic subcontinuum. This is a somewhat stronger result than the known equivalence of hereditary local connectedness and hereditary aposyndesis.

Mathematical Subject Classification
Primary: 54.55
Received: 9 January 1969
Published: 1 October 1969
Eldon Jon Vought