Balanced ideals in cozero parts of frames

Abstract

We study balanced filters and balanced z-filters considered by Carlson in [20] and [21] in topological spaces. We consider closed filters which are open-generated and open filters which are closed-generated. We show that a closed filter is open-generated precisely if it is a minimal balanced closed filter and that an open filter is closed-generated precisely when it is a minimal balanced open filter. For a completely regular topological space X, we study balanced z-filters and show that there is a one-to-one correspondence between the nonempty closed sets of βX and the balanced z-filter on X. By dualising closed filters we obtain ideals which then enables us to put some of the results in the context of frames. Dube in [28] has shown that a frame is normal if and only if its closed-generated filters are precisely the stably closed-generated ones. By dualisation we show that a frame is extremally disconnected if and only if its open-generated ideals are precisely the stably open-generated ones. We show that there is one-to-one correspondence between points of βL and the balanced ideals of Coz L. Furthermore we study nearness frames and show that the locally finite nearness frames strictly contain the Pervin nearness frames and the two coincide if the locally finite nearness frames are totally bounded. For perfect extension h : M → L of L, we show that a point p of M is a remote point if and only if Ip = {a ∈ L | h∗(a) ≤ p}.