Decompositions that destroy simple connectivity.

*(English)*Zbl 0608.57012The author surely did more than any other individual to define the direction of set theoretic topology in the middle years of this century, especially in the area of decompositions of manifolds. He raised questions, developed techniques, and created examples that have stimulated several academic generations of topologists.

Prior to his death (in April, 1986), Bing recognized that he had never published a proof of a statement of his that appeared in print in 1957. This paper rectifies that omission by proving that the decomposition space whose only nondegenerate element is a solenoid is not simply connected. The proof is ”vintage Bing”, using the same kind of clarity of vision that informed many of his arguments, but requiring close attention to details. Bing wanted his readers to understand something of the genesis of his ideas and how they developed into final form. There is even the expected unanswered question suggesting related possible areas of investigation.

Prior to his death (in April, 1986), Bing recognized that he had never published a proof of a statement of his that appeared in print in 1957. This paper rectifies that omission by proving that the decomposition space whose only nondegenerate element is a solenoid is not simply connected. The proof is ”vintage Bing”, using the same kind of clarity of vision that informed many of his arguments, but requiring close attention to details. Bing wanted his readers to understand something of the genesis of his ideas and how they developed into final form. There is even the expected unanswered question suggesting related possible areas of investigation.

Reviewer: L.O.Cannon

##### MSC:

57N12 | Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |

57M40 | Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |

54B15 | Quotient spaces, decompositions in general topology |