From a locally compact space X one construct its Cech-Stone remainder X*=beta X minus X. We analyze the problem on whether X* and Y* can be homeomorphic for different spaces X and Y. In the 0-dimensional case, a solution to this problem has been proved to be independent of ZFC, by the work of Parovicenko, Farah, Dow-Hart and Farah-McKenney among others.

We prove, under PFA, the strongest possible rigidity result: for metrizable X and Y, we prove that X* is homeomorphic to Y* only if X and Y are homeomorphic modulo compact subsets. We also show that every homeomorphism X* --> Y* lifts to an homeomorphism between cocompact subsets of X and Y.