## Indecomposable homogeneous plane continua are hereditarily indecomposable

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- by Charles L. Hagopian PDF
- Trans. Amer. Math. Soc.
**224**(1976), 339-350 Request permission

## Abstract:

F. Burton Jones [7] proved that every decomposable homogeneous plane continuum is either a simple closed curve or a circle of homogeneous nonseparating plane continua. Recently the author [5] showed that no subcontinuum of an indecomposable homogeneous plane continuum is hereditarily decomposable. It follows from these results that every homogeneous plane continuum that has a hereditarily decomposable subcontinuum is a simple closed curve. In this paper we prove that no subcontinuum of an indecomposable homogeneous plane continuum is decomposable. Consequently every homogeneous nonseparating plane continuum is hereditarily indecomposable. Parts of our proof follow one of R. H. Bing’s arguments [2]. At the Auburn Topology Conference in 1969, Professor Jones [8] outlined an argument for this theorem and stated that the details would be supplied later. However, those details have not appeared.## References

- R. H. Bing,
*A homogeneous indecomposable plane continuum*, Duke Math. J.**15**(1948), 729–742. MR**27144** - R. H. Bing,
*A simple closed curve is the only homogeneous bounded plane continuum that contains an arc*, Canadian J. Math.**12**(1960), 209–230. MR**111001**, DOI 10.4153/CJM-1960-018-x - Edward G. Effros,
*Transformation groups and $C^{\ast }$-algebras*, Ann. of Math. (2)**81**(1965), 38–55. MR**174987**, DOI 10.2307/1970381 - G. R. Gordh Jr.,
*On homogeneous hereditarily unicoherent continua*, Proc. Amer. Math. Soc.**51**(1975), 198–202. MR**375254**, DOI 10.1090/S0002-9939-1975-0375254-6 - Charles L. Hagopian,
*Homogeneous plane continua*, Houston J. Math.**1**(1975), 35–41. MR**383369** - F. Burton Jones,
*Certain homogeneous unicoherent indecomposable continua*, Proc. Amer. Math. Soc.**2**(1951), 855–859. MR**45372**, DOI 10.1090/S0002-9939-1951-0045372-4 - F. Burton Jones,
*On a certain type of homogeneous plane continuum*, Proc. Amer. Math. Soc.**6**(1955), 735–740. MR**71761**, DOI 10.1090/S0002-9939-1955-0071761-1
—, - K. Kuratowski,
*Topology. Vol. II*, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR**0259835**
S. Mazurkiewicz, - Edwin E. Moise,
*A note on the pseudo-arc*, Trans. Amer. Math. Soc.**67**(1949), 57–58. MR**33023**, DOI 10.1090/S0002-9947-1949-0033023-X - R. L. Moore,
*Concerning upper semi-continuous collections of continua*, Trans. Amer. Math. Soc.**27**(1925), no. 4, 416–428. MR**1501320**, DOI 10.1090/S0002-9947-1925-1501320-8 - R. L. Moore,
*Foundations of point set theory*, Revised edition, American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR**0150722** - E. S. Thomas Jr.,
*Monotone decompositions of irreducible continua*, Rozprawy Mat.**50**(1966), 74. MR**196721**

*Homogeneous plane continua*, Proc. Auburn Topology Conf., Auburn Univ.,

*Sur les points accessibles des continus indecomposables*, Fund. Math.

**14**(1929), 107-115.

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**224**(1976), 339-350 - MSC: Primary 54F20
- DOI: https://doi.org/10.1090/S0002-9947-1976-0420572-1
- MathSciNet review: 0420572