Schlais' theorem extends to connected plane continua

Author:
Charles L. Hagopian

Journal:
Proc. Amer. Math. Soc. **40** (1973), 265-267

MSC:
Primary 54F15; Secondary 54F20

MathSciNet review:
0317297

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We call a nondegenerate metric space that is compact and connected a continuum. For each point of a continuum , F. B. Jones [**2**] defines to be the set consisting of all points of such that is not aposyndetic at with respect to . H. E. Schlais [**4**] proved that if is a hereditarily decomposable continuum, then for each point of , no nonempty open set in is contained in . A continuum is said to be connected if any two of its points can be joined by a hereditarily decomposable continuum in . Here we prove that if is a connected plane continuum, then for each point of , the set does not contain a nonempty open subset of .

**[1]**Charles L. Hagopian,*𝜆 connected plane continua*, Trans. Amer. Math. Soc.**191**(1974), 277–287. MR**0341435**, 10.1090/S0002-9947-1974-0341435-4**[2]**F. Burton Jones,*Concerning non-aposyndetic continua*, Amer. J. Math.**70**(1948), 403–413. MR**0025161****[3]**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****[4]**H. E. Schlais,*Non-aposyndesis and non-hereditary decomposability*, Pacific J. Math.**45**(1973), 643–652. MR**0317298**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54F15,
54F20

Retrieve articles in all journals with MSC: 54F15, 54F20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0317297-2

Keywords:
Hereditarily decomposable continua,
lambda connected plane continua,
Jones functions ,
arcwise connectivity,
Schlais theorem

Article copyright:
© Copyright 1973
American Mathematical Society