A Pointfree Study of Bases For Spaces Of Minimal Prime Ideals

Abstract

Let C (X) denote the ring of all real-valued continuous functions on a topological space X, and mX its space of minimal prime ideals with the hull-kernel topology. It has a base consisting of clopen sets and which contains the closure of the union of any sequence of its members. In the paper Spaces with a pretty base ( J. Pure and Appl. Algebra 70 (1991), 81-87), spaces with a base with these properties are studied and are shown to have almost all of the known properties of the space mX including the fact that if it is a weakly Lindel¨ of space, then it is basically disconnected. In the present paper, analogous results are derived in a pointfree context in which topological spaces are replaced by frames. In some cases, we are able to obtain more general results with simpler proofs.

Keywords:

• PRETTY BASE
• MINIMAL PRIME IDEAL
• HULL-KERNEL TOPOLOGY
• QUOTIENT FRAME
• NUCLEUS
• PSEUDOCOMPLEMENT
• SEPARATED
• WEAKLY LINDELOF
• BASICALLY DISCONNECTED
• P-FRAME
• COCOMPACT
• QUASI-COMPLEMENTED