Vol. 34, No. 1, 1970

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On generalized forms of aposyndesis

Charles Lemuel Hagopian

Vol. 34 (1970), No. 1, 97–108

If a point set is both connected and closed it is called a continuum. The structure of a nonlocally connected continuum can be described in terms of its aposyndetic properties. In this paper various forms of continuum aposyndesis, that is, aposyndesis with respect to subcontinua, are considered. It is shown that the presence of any of these forms of aposyndesis in a compact metric continuum which is totally nonconnected im kleinen (not connected im kleinen at any point) insures nonsemi-local-connectedness on a dense open subset of the continuum and the set of weak cut points in each open subset of the continuum has cardinality at least c1 A weak cut point theorem for compact plane continua is established. An example is given which indicates that this result does not hold in Euclidean 3-space. Near aposyndesis, a generalization of aposyndesis, is introduced. It is shown that the presence of this property in a totally nonaposyndetic, separable, metric continuum implies the existence of uncountably many weak cut points.

Mathematical Subject Classification
Primary: 54.55
Received: 19 January 1970
Revised: 24 February 1970
Published: 1 July 1970
Charles Lemuel Hagopian