NEARNESSES AND To - EXTENSIONS OF TOPOLOGICAL SPACES

ABSTRACf. In [3], Reed establishes a bijection between the (equivalence classes of) principal Tt-extensions of a topological space X and the compatible, cluster-generated, Lodato nearnesses on X. We extend Reed's result to the To case by obtaining a one-to-one correspondence between the principal To-extensions of a space X and the collections of sets (called "t-grill sets") which generate a certain class of nearnesses which we call "t-bunch generated" nearnesses. This correspondence specializes to principal To-compactifications. Finally, we show that there is a bijection between these t-grill sets and the filter systems of Thron [5], and that the corresponding extensions are equivalent.