On a problem of Čech

Assuming CH (but we have stronger results) we partially solve a problem posed by E. ?ech. Kuratowski gave the axioms which a topological closure operator, ?, must satisfy. If we do not ask that ? be idempotent (i.e. that for all $\text{X,(?(X))=?(X)}$, then ? is known as a closure operator. E. ?ech asked if there is a nontrivial closure operator which is onto, that is, for which in some sense every subset of our ‘space’ is ‘closed’. We build such functions. The problem is also well motivated when presented in purely set theoretic terms.