SOME CLASSIFICATIONS OF SUBRINGS OF THE RING OF CONTINUOUS FUNCTIONS

Abstract

In this work, for a topological space \( (X,\tau) \) and the ring of functions \( C(X,\tau)=\{ f \mid f:(X,\tau)\to(\mathbb{R},\tau_{st}) \text{ and } f \text{ is continuous} \} \), where \( \tau_{st} \) is the standard topology on the real line, for a closed, sublattice and subring \( \mathcal{A} \) of \( C(X,\tau) \) we investigate sufficient conditions under which is not possible to find a topology \( \tau' \subset \tau \) on \( X \) such that \( \mathcal{A}=C(X,\tau') \).