Images and Open Subspaces of SV Spaces

Abstract

A topological space is finitely an F-space if its Stone–Čech compactification is a union of finitely many closed F-spaces and a space is SV if C(X) has the property that C(X)/P is a valuation domain for each prime ring ideal P of C(X). This article studies the images under open continuous functions and the open subspaces of spaces that are finitely an F-space or are SV. It is shown that an open continuous image of a compact space that is finitely an F-space is finitely an F-space and an open continuous image of certain SV spaces is SV. Also, it is shown cozerosets, but not necessarily open sets, of SV spaces are SV spaces and a similar situation holds for spaces that are finitely an F-space.

Key Words:

• Finite rank
• F-rings
• F-space
• Open functions
• SV-ring
• SV space
• Rings of continuous functions

2000 Mathematics Subject Classification:

• Primary 06F25, 54C40, 54G05
• Secondary 54C10

Notes

Communicated by I. Swanson.