Another homogeneous plane continuum

Abstract

1. History of problem. A set X is homogeneous if for each pair of points x, y of X there is a homeomorphism of X onto itself that takes x into y. In 1920 Knaster and Kuratowski [21] raised the following question: If a nondegenerate bounded plane continuum is homogeneous, is it necessarily a simple closed curve? In 1922, Knaster [20] described a hereditarily indecomposable plane continuum. It is reported that he suspected that this continuum had other interesting properties and suggested to his students the problem of discovering if this Knaster continuum (as it came to be called) had the property possessed by an arc of being topologically equivalent to each of its nondegenerate subcontinua. This Knaster continuum is homogeneous and furnishes a counterexample to an affirmative answer of the above question, but this was not discovered until 1951. A partial affirmative solution was given to the question in 1924 when Mazurkiewicz [22] showed that the bounded nondegenerate homogeneous plane continuum is a simple closed curve if it is locally connected. This result was improved in 1951 when Cohen [12] showed that the continuum is a simple closed curve if it either is arcwise connected or contains a simple closed curve. A false affirmative solution was announced [25] in 1937. (Of course, at the time, it was not known that the solution was false-this only developed eleven years later when a counter-example was given.) This false solution was extended [II] in 1944 when an attempt was made to classify all homogeneous bounded closed plane sets. That a pseudo-arc is homogeneous was shown first by Bing [3] in 1948 and shortly thereafter by essentially the same methods by Moise [24]. Both of these proofs made use of the description of the pseudo-arc given by Moise [23] to show that a pseudo-arc is topologically equivalent to each of its nondegenerate subcontinua. In 1951 Bing [4] discovered that the pseudo-arc described by Moise in 1948 is actually topologically equivalent to the continuum Knaster described twenty-six years earlier by different methods. In fact, it was shown that any two nondegenerate snake-like hereditarily indecomposable continua are topologically equivalent. Also, it was shown that in the category sense, most