Monotone decompositions of Hausdorff continua

Vought, Eldon J.

doi:10.1090/S0002-9939-1976-0410693-7

Monotone decompositions of Hausdorff continua
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by Eldon J. Vought

Proc. Amer. Math. Soc. 56 (1976), 371-376

DOI: https://doi.org/10.1090/S0002-9939-1976-0410693-7

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Abstract:

A monotone, upper semicontinuous decomposition of a compact, Hausdorff continuum is admissible if the layers (tranches) of the irreducible subcontinua of $M$ are contained in the elements of the decomposition. It is proved that the quotient space of an admissible decomposition is hereditarily arcwise connected and that every continuum $M$ has a unique, minimal admissible decomposition $\mathcal {A}$. For hereditarily unicoherent continua $\mathcal {A}$ is also the unique, minimal decomposition with respect to the property of having an arcwise connected quotient space. A second monotone, upper semicontinuous decomposition $\mathcal {G}$ is constructed for hereditarily unicoherent continua that is the unique minimal decomposition with respect to having a semiaposyndetic quotient space. Then $\mathcal {G}$ refines $\mathcal {G}$ and $\mathcal {G}$ refines the unique, minimal decomposition $\mathcal {L}$ of FitzGerald and Swingle with respect to the property of having a locally connected quotient space (for hereditarily unicoherent continua).

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