Vol. 19, No. 2, 1966

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A theorem on one-to-one mappings

Edwin Duda

Vol. 19 (1966), No. 2, 253–257

Let X be a locally connected generalized continuum with the property that the complement of each compact set has only one nonconditionally compact component. The author proves the following theorem. If f is a one-to-one mapping of X onto Euclidean 2-space, then f is a homeomorphism.

An example of K. Whyburn implies that if f is a one-to-one mapping of X onto Euclidean n-space (n 3), then X can have many nice properties any yet f need not be a homeomorphism. However the complement of a compact set in the domain space of his example may have more than one nonconditionally compact component.

It is interesting to note that a characterization of closed 2-cells in the plane is obtained in the course of proving the theorem.

Mathematical Subject Classification
Primary: 54.60
Received: 6 August 1965
Published: 1 November 1966
Edwin Duda