Monotone decompositions of Hausdorff continua

Vought, Eldon J.

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

Monotone decompositions of Hausdorff continua
HTML articles powered by AMS MathViewer

by Eldon J. Vought PDF

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

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).

References

R. H. Bing, Some characterizations of arcs and simple closed curves, Amer. J. Math. 70 (1948), 497–506. MR 25722, DOI 10.2307/2372193
J. J. Charatonik, On decompositions of continua, Fund. Math. 79 (1973), no. 2, 113–130. MR 321034, DOI 10.4064/fm-79-2-113-130
J. J. Charatonik, On decompositions of $\lambda$-dendroids, Fund. Math. 67 (1970), 15–30. MR 261556, DOI 10.4064/fm-67-1-15-30
R. W. FitzGerald and P. M. Swingle, Core decomposition of continua, Fund. Math. 61 (1967), 33–50. MR 224063, DOI 10.4064/fm-61-1-33-50
G. R. Gordh Jr., Concerning closed quasi-orders on hereditarily unicoherent continua, Fund. Math. 78 (1973), no. 1, 61–73. MR 322835, DOI 10.4064/fm-78-1-61-73
G. R. Gordh Jr., Monotone decompositions of irreducible Hausdorff continua, Pacific J. Math. 36 (1971), 647–658. MR 281163
F. Burton Jones, Aposyndetic continua and certain boundary problems, Amer. J. Math. 63 (1941), 545–553. MR 4771, DOI 10.2307/2371367
F. Burton Jones, Concerning non-aposyndetic continua, Amer. J. Math. 70 (1948), 403–413. MR 25161, DOI 10.2307/2372339
Louis F. McAuley, An atomic decomposition of continua into aposyndetic continua, Trans. Amer. Math. Soc. 88 (1958), 1–11. MR 124033, DOI 10.1090/S0002-9947-1958-0124033-8