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/