Topology Proceedings 6 (1981) pp. 135-158: WHAT RESULTS ARE VALID ON CLOSURE SPACES

concept of nearness (contiguity, proximity) between families of sets (finite families, pairs). Again F. Riesz in 1908 was the first to explore this path by defining a "Verkettung" between pairs of sets. He did not require A n B ~ ¢ ~ A is "near" B. Today one generally makes this assumption and thus obtains c (A) = [x: [x] is "near" A]. The assumptions CI , C2 , C3 follow from requirements usually imposed on near structures. It thus becomes desirable to study structures (X,c), where c is only required to satisfy the axioms C C C l , 2 , 3 . Such structures were called closupe spaces by Cech (1966) and were investigated by him. Since that time further results have been obtained and it is now clear that topological spaces do not constitute a natural boundary for the validity of theorems but that many results can be extended to closure spaces. We give TOPOLOGY PROCEEDINGS Volume 6 1981 137 a brief survey of recent investigations of weak idem­ potency, separation axioms, types of compactness, exten­ sions (in particular principal extensions), and a corre­ spondence between certain near structures and certain com­ pactifications, among others. We shall give proofs only for those results which have not previously been published or submitted for publication. There are two articles: Chattopadhyay and Thron [1977] and Chattopadhyay, Nj~stad and Thron [submitted] which we shall cite frequently. We shall refer to the first as CT and the second as CNT. 2. Grills and Basic Properties of Closure Spaces In our treatment of closure spaces grills play a key role. A family § c ~(X) is called a grill if ¢ ~ §, B :::> A E § ~ B E §, A U B E § ~ A E § or B E §. We use the following notation f(X) is the set of all grills on X, ~(X) is the collection of all filters on X, and ~(X) is the set of all ultrafilters on X. For all § E f(X) one defines yt = [U: U E [2 (X), U c y] . §t is thus a subset of ~(X). The mapping d: ~(X) U f(X) ~ ~(X) U f(X) given by d (#) [B: B n H :I ¢ VH E #] provides a 1 1 mapping from ~(X) to f(X). The following simple relations hold.