Sierpinski has shown (Wacław Sierpiński Sur une condition pour qu'un continu soit une courbe jordanienne, Fundamenta Mathematicae I (1920), pp. 44-60) that in order that a closed and connected set of points M should be a continuous curve it is necessary and sufficient that, for every positive number ϵ, the connected point-set M should be the sum of a finite number of closed and connected point-sets each of diameter less than ϵ. It follows that, as applied to point-sets which are closed, bounded and connected, this property is equivalent to that of connectedness "im kleinen". The purpose of the present paper is to make a further study of these two properties (or rather suitable modifications of these properties) especially as applied to sets which are not necessarily closed.
@article{bwmeta1.element.bwnjournal-article-fmv3i1p23bwm,
author = {R. Moore},
title = {Concerning connectedness im kleinen and a related property},
journal = {Fundamenta Mathematicae},
volume = {3},
year = {1922},
pages = {232-237},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv3i1p23bwm}
}
Moore, R. Concerning connectedness im kleinen and a related property. Fundamenta Mathematicae, Tome 3 (1922) pp. 232-237. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv3i1p23bwm/