The purpose of the present note is to show that no connected point set can have more than one point such that when it is removed, the reminder is totally disconnected.
@article{bwmeta1.element.bwnjournal-article-fmv3i1p24bwm, author = {John Kline}, title = {A theorem concerning connected point sets}, journal = {Fundamenta Mathematicae}, volume = {3}, year = {1922}, pages = {238-239}, zbl = {48.0210.01}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv3i1p24bwm} }
Kline, John. A theorem concerning connected point sets. Fundamenta Mathematicae, Tome 3 (1922) pp. 238-239. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv3i1p24bwm/