Schlais’ theorem extends to $\lambda$ connected plane continua
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- by Charles L. Hagopian PDF
- Proc. Amer. Math. Soc. 40 (1973), 265-267 Request permission
Abstract:
We call a nondegenerate metric space that is compact and connected a continuum. For each point $x$ of a continuum $M$, F. B. Jones [2] defines $K(x)$ to be the set consisting of all points $y$ of $M$ such that $M$ is not aposyndetic at $x$ with respect to $y$. H. E. Schlais [4] proved that if $M$ is a hereditarily decomposable continuum, then for each point $x$ of $M$, no nonempty open set in $M$ is contained in $K(x)$. A continuum $M$ is said to be $\lambda$ connected if any two of its points can be joined by a hereditarily decomposable continuum in $M$. Here we prove that if $M$ is a $\lambda$ connected plane continuum, then for each point $x$ of $M$, the set $K(x)$ does not contain a nonempty open subset of $M$.References
- Charles L. Hagopian, $\lambda$ connected plane continua, Trans. Amer. Math. Soc. 191 (1974), 277–287. MR 341435, DOI 10.1090/S0002-9947-1974-0341435-4
- F. Burton Jones, Concerning non-aposyndetic continua, Amer. J. Math. 70 (1948), 403–413. MR 25161, DOI 10.2307/2372339
- 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
- H. E. Schlais, Non-aposyndesis and non-hereditary decomposability, Pacific J. Math. 45 (1973), 643–652. MR 317298, DOI 10.2140/pjm.1973.45.643
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 40 (1973), 265-267
- MSC: Primary 54F15; Secondary 54F20
- DOI: https://doi.org/10.1090/S0002-9939-1973-0317297-2
- MathSciNet review: 0317297