Fixed Points and Noetherian Topologies

Abstract

This paper provides a canonical construction of a Noetherian least fixed point topology. While such least fixed point are not Noetherian in general, we prove that under a mild assumption, one can use a topological minimal bad sequence argument to prove that they are. We then apply this fixed point theorem to rebuild known Noetherian topologies with a uniform proof. In the case of spaces that are defined inductively (such as finite words and finite trees), we provide a uniform definition of a divisibility topology using our fixed point theorem. We then prove that the divisibility topology is a generalisation of the divisibility preorder introduced by Hasegawa in the case of well-quasi-orders.