ALMOST LINDELÖF FRAMES

Generalizing $Lindel{\ddot{o}}f$ frames and almost compact frames, we introduce a concept of almost $Lindel{\ddot{o}}f$ frames. Using a concept of ${\delta}$-filters on frames, we characterize almost $Lindel{\ddot{o}}f$ frames and then have their permanence properties. We also show that almost $Lindel{\ddot{o}}f$ regular $D({\aleph}_1)$ frames are exactly $Lindel{\ddot{o}}f$ frames. Finally we construct an almost $Lindel{\ddot{o}}fication$ of a frame L via the simple extension of L associated with the set of all ${\delta}$-filters F on L with ${\bigvee}\{x^*{\mid}x{\in}F\}=e$.