Some new characterizations of pointfree pseudocompactness

Abstract

By [3], a frame L is pseudocompact iff every ≺≺-sequence in L joining to the top terminates. Here it is shown, for any completely regular L, that pseudocompactness is also equivalent to (i) the analogous condition for ≺-sequences, (ii) the countable almost compactness of L, (iii) the almost compactness of CozL as a σ-frame and (iv) the condition that every countably based proper filter in L clusters. Further we establish the zero-dimensional counterparts of the above, concerning the integer valued notion of pseudocompactness. Finally, we add to this a characterization of pseudocompactness in terms of uniformities.