Let X be a compact metric space and let |A| denote the cardinality of a set A. We prove that if is a homeomorphism and , then for all > 0 there is such that |A| = 4 and for all there are , , such that dist(x, y) < . An observer that can only distinguish two points if their distance is grater than , for sure will say that A has at most 3 points even knowing every iterate of A and that f is a homeomorphism. We show that for hyperexpansive homeomorphisms the same -observer will not fail about the cardinality of A if we start with |A| = 3 instead of 4. Generalizations of this problem are considered via what we call (m, n)-expansiveness.

topological dynamics;expansive homeomorphisms;

English