The widths of simple trees

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1974

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Abstract

This paper is concerned with the concept of width, originally defined for tree-like continua by C. E. Burgess in his paper. Collections and sequences of continua in the plane, II, Pacific Journal of Mathematics, v. 11(1961). Dr. A. Lelek later redefined width as applied to simple trees. The two definitions are shown to be equivalent, then some results are presented which were obtained using the Lelek definition. A primary concern was determining the relationship between the width of a simple tree and the widths of its subcontinua which are simple triods - called subtriods. It is shown that, given a simple tree X of width w(X), there exists a width factor, m(X), such that, for some subtriod T of X, w(T) > m(X)*w(X). Furthermore, m(X) is topologically invariant. A second theorem establishes that if X is a simple tree of w(X) > [element of] > 0, and X contains exactly one ramification point (that is, X is an n-od) such that the order of the ramification point is n = 4, 5, or 6, then X contains a subtriod T such that w(T) > ([element of]/2). Finally, there are some examples to show that it is not true that if X is a simple tree such that w(X) > [element of] > 0, then X contains a subtriod T such that w(T) > [element of].

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