A note on weakly Lindelöf frames

Abstract

The notion of σ*-properness of a subset of a frame is introduced. Using this notion, we give necessary and sufficient conditions for a frame to be weakly Lindelöf . We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf . For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P-frame is realcompact. A new characterization of weakly Lindelöf  frames in terms of neighbourhood strongly divisible ideals of RL is provided. The closed ideals of RL equipped with the uniform topology are applied to describe weakly Lindelöf  frames.

Mathematics Subject Classification (2010): Primary: 06D22; Secondary: 54D20, 54D60.

Keywords: Frame, weakly Lindelöf  frame, weakly realcompact frame, σ*-proper, neighbourhood strongly divisible ideal, uniformly closed ideal