A plane continuum is constructed which has span zero but is not chainable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-2-3,
author = {L. C. Hoehn},
title = {A non-chainable plane continuum with span zero},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {149-174},
zbl = {1238.54013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-2-3}
}
L. C. Hoehn. A non-chainable plane continuum with span zero. Fundamenta Mathematicae, Tome 215 (2011) pp. 149-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-2-3/