On infinite versions of the prisoner problem

We investigate some versions of the famous 100 prisoner problem for the infinite case, where there are infinitely many prisoners and infinitely many boxes with labels. In this case, many questions can be asked about the admissible steps of the prisoners, the constraints they have to follow and also about the releasing conditions. We will present and analyze many versions and cases. In the infinite case, the solutions and methods require mainly analysis rather than combinatorics.