Classification of all hereditarily indecomposable circularly chainable continua

Fearnley, Lawrence

doi:10.1090/S0002-9947-1972-0296903-9

Classification of all hereditarily indecomposable circularly chainable continua
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by Lawrence Fearnley

Trans. Amer. Math. Soc. 168 (1972), 387-401

DOI: https://doi.org/10.1090/S0002-9947-1972-0296903-9

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Abstract:

In a recent paper the author has established an affirmative solution to a well-known and previously unsolved problem raised by R. H. Bing in 1951 concerning whether or not the pseudo-circle is topologically unique. Now in this present paper, as the natural culmination of the study initiated in the earlier paper, complete topological and mapping classification theorems are established for all hereditarily indecomposable circularly chainable continua. The principal topological classification results of this paper are the theorems that hereditarily indecomposable circularly chainable continua are characterized set-theoretically by their equivalence classes of fundamental sequences and are characterized algebraic-topologically by their Čech cohomology groups. These topological classification theorems are then used in establishing mapping classification theorems for all hereditarily indecomposable circularly chainable continua and in proving that the mapping hierarchy of hereditarily indecomposable circularly chainable continua constitutes a lattice. Among the consequences of the foregoing primary results of the paper are the additional theorems that two hereditarily indecomposable circularly chainable continua are topologically equivalent if and only if each of them is a continuous image of the other, and that every $k$-adic pseudo-solenoid is topologically unique.

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