"In 1920, B. Knaster and C. Kuratowski raised the question of whether each homogeneous plane continuum is a simple closed curve. In 1921, S. Mazurkiewicz raised the question of whether each subcontinuum of Euclidean n-space which is homeomorphic to each of its subcontinua is necessarily an arc. In that same year, B. Knaster and C. Kuratowski raised the question of whether there exists a nondegenerate hereditarily indecomposable continuum.
The third question was answered in the affirmative in 1922 by B. Knaster, when he constructed a nondegenerate hereditarily indecomposable subcontinuum of the plane.
The second question was answered in 1947 by E. Moise. The continuum that Moise constructed to answer the second question is a nondegenerate hereditarily indecomposable continuum that lies in the plane, and which has the interesting property that it is homeomorphic to each of its nondegenerate subcontinua but is not an arc.
In 1948 R.H. Bing answered the first question. The continuum that Bing constructed is lies in the plane and is hereditarily indecomposable. It was also shown by Bing that his continuum is homogeneous but not a simple closed curve.
Each of the examples constructed to answer the above questions is a nondegenerate, hereditarily indecomposable, and chainable continuum. It is the intent of this paper to show that if M and P are nondegenerate, hereditarily indecomposable chainable continua, then M and Pare homeomorphic. Moreover, we shall prove that if M is a nondegenerate, hereditarily indecomposable chainable continuum, then Mis homogeneous and hereditarily equivalent"--Abstract, p. iii
Robert P. Roe
William T. Ingram
Daniel J. Okunbor
Mathematics and Statistics
M.S. in Applied Mathematics
University of Missouri--Rolla
v, 56 pages
© 1997 Thomas John Kacvinsky, All rights reserved.
Thesis - Open Access
Print OCLC #
Kacvinsky, Thomas John, "Some properties of hereditarily indecomposable chainable continua" (1997). Masters Theses. 1706.