The purpose of this paper is to prove: Theorem: Suppose M is a closed connected set containing more than one point such that if g is any connected subset of M, then M-g is connected. Under these conditions M is a simple closed curve. Theorem: If M is an unbounded closed connected set which remains connected upon the removal of any unbounded connected proper subset, then M is either an open curve, a ray of an open curve or a simple closed curve J plus OP, a ray of an open curve which has O and only O in common with J. Theorem: Suppose M is an unbounded closed connected set such that if g is any bounded connected subset of M, then M-g is connected. Then M is not a continuous curve.
@article{bwmeta1.element.bwnjournal-article-fmv5i1p2bwm,
author = {John Kline},
title = {Closed connected sets which remain connected upon the removal of certain, connected subsets},
journal = {Fundamenta Mathematicae},
volume = {6},
year = {1924},
pages = {3-10},
zbl = {50.0138.01},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv5i1p2bwm}
}
Kline, John. Closed connected sets which remain connected upon the removal of certain, connected subsets. Fundamenta Mathematicae, Tome 6 (1924) pp. 3-10. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv5i1p2bwm/