The Pseudo-circle Is Not Homogeneous

In the first two volumes of Fundamenta Mathematica, Knaster and Kuratowski raised the following two questions [15], [16]: (1) If a nondegenerate, bounded plane continuum is homogeneous, is it necessarily a simple closed curve? (2) Does there exist a continuum each subcontinuum of which is indecomposable? Although Knaster settled the second question in 1922 [14], it was to remain until 1948 for the interrelationship of the two questions to become apparent. In that year Moise [18] described an indecomposable, chainable plane continuum which was homeomorphic to each of its proper subcontinua, and Bing [1 ] showed that this continuum was homogeneous. Bing [2] later showed that this continuum, which Moise called a pseudo-arc, was homeomorphic to Knaster's continuum. In 1951 Bing [2] showed the existence of uncountably many topologically distinct, hereditarily indecomposable, plane continua. The continuum which he used as a tool in his construction was itself a promising candidate for inclusion in the class of homogeneous, hereditarily indecomposable, plane continua. This circularly chainable continuum had the property that, while it was topologically different from a pseudo-arc, every proper subcontinuum of it was a pseudo-arc ; hence it seemed to combine some of the properties of the pseudo-arc and the simple closed curve. Such continua were later referred to as "pseudo-circles" and were discussed by F. B. Jones in [11]. In [20] the author discussed hereditarily indecomposable, circularly chainable continua in general and constructed uncountably many such continua, one of which had the additional property that it could be mapped continuously onto any circularly chainable continuum. The purpose of this paper is to prove that the pseudo-arc is the only homogeneous, hereditarily indecomposable, circularly chainable continuum (Theorem 2). It will first be shown that the pseudo-circle is not homogeneous (Theorem 1) and the methods will be adapted to prove the more general theorem. C. E. Burgess [11 ] has asked whether each continuum, each proper subcontinuum of which is homogeneous, is itself homogeneous. Since each proper subcontinuum of a pseudo-circle is a pseudo-arc and hence is homogeneous, this paper answers Burgess' question in the negative. In this paper a continuum is a nondegenerate, compact, connected subset of a metric space. A map or a mapping is a continuous, single-valued function. A