A theorem on continuous decompositions of the plane into nonseparating continua

Smith, Michel

doi:10.1090/S0002-9939-1976-0415558-2

A theorem on continuous decompositions of the plane into nonseparating continua
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Proc. Amer. Math. Soc. 55 (1976), 221-222 Request permission

Abstract:

E. Dyer [2] proved that there is no continuous decomposition of a compact irreducible continuum into decomposable continua which is an arc with respect to its elements. The author extends Dyer’s result to the plane. Consider a continuous decomposition of the plane into nonseparating compact continua. R. L. Moore [6] has shown that the decomposition space is homeomorphic to the plane. Using Moore’s result it is shown that the union of the elements of each arc in the decomposition space is an irreducible continuum. It follows then, from Dyer’s result, that there is no continuous decomposition of the plane into nonseparating compact decomposable continua.

References

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There does not exist an upper semi-continuous decomposition of

into arcs

2