Topological types of finitely-𝐶⁰-𝐾-determined map-germs

Abstract

In this article, we investigate the following two problems Problem 1. Is finite- C 0 - K {C^0}{\text {-}}K -determinacy a topological invariant among analytic map-germs? Problem 2. Do the topological types of all finitely- C 0 - K {C^0}{\text {-}}K -determined map-germs have topological moduli, i.e. do they have infinitely many topological types with the cardinal number of continuum? Problem 1 1 is solved affirmatively in the complex case. Problem 2 2 is solved negatively in the complex case; and affirmatively in the real case.