Zero span continua

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1981
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Abstract

This dissertation is concerned with the span of continua and with continua in class W. The emphasis of this study is on the class of continua which have zero span. The relationship between this class and class W is studied. The author introduces in this dissertation the notion of the symmetric span of a continuum by modifying Lelek's origional definition. Continua with zero span have zero symmetric span but the converse does not hold. The author proves that the continua with zero symmetric span are in class W. Lelek defined the semispan of a continuum M by relaxing a condition in the definition of span. Lelek gives an example of a continuum for which the span and semispan are different. The author shows that if every subcontinuum of the continuum M is in class W, then the span and semispan of M are the same. This result and the previous one are then applied to show that the span of a continuum is zero if and only if the semispan is zero. This has the corollary that continua with zero span have the fixed point property. A second corollary of this theorem generalizes a recent result of Duda and Kell to the following: If H and K are continua with zero span which intersect, H n K is connected, and H u K is atriodic then H u K has zero span. Next, decomposable, atriodic continua are studied. After preliminary lemmas. It is shown that if H and K are continua in class W which intersect, H n K is connected, and H u K is atriodic then H u K is in class W. Finally certain homogeneous continua are studied. It is shown that homogeneous, almost chainable continua have zero span.

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Keywords
Metric spaces, Continuity
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