A nontrivial surjective Čech closure function is constructed in ZFC.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-2-6,
author = {Fred Galvin and Petr Simon},
title = {A \v Cech function in ZFC},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {181-188},
zbl = {1118.54002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-2-6}
}
Fred Galvin; Petr Simon. A Čech function in ZFC. Fundamenta Mathematicae, Tome 193 (2007) pp. 181-188. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-2-6/