NONSTANDARD EXPANSIVENESS

The paper’s Theorem 2 states precisely that a homeomorphism on a compact metric space is nonstandard expansive if and only if it is expansive and has no doubly asymptotic points, leveraging the NSA characterization of asymptotic pairs (Proposition 3.3) and a nearstandard/standard-part argument to obtain separation at an infinite time (with a possible constant degradation, e.g., to c/2) . Expansiveness in the (⇒) direction follows from the earlier equivalence between nonstandard expansiveness and having infinitely many separating times (Theorem 1) . The candidate solution proves the same equivalence by transfer and an internal-set underflow/overspill argument, preserving the same constant and reaching a contradiction via compactness and accumulation points. Both arguments are logically sound and establish the same result by different methods.