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Abstract. Gagrat, Naimpally, and Thron together have shown that separated Lodato proximities yield 7",-compactifications, and conversely. This correspondence is not 1-1, since nonequivalent compactifications can induce the same proximity. Herrlich has shown that if the concept of proximity is replaced by that of nearness then all principal (or strict) 7",-extensions can be accounted for. (In general there are many nearnesses compatible with a given proximity.) In this paper we obtain a 1-1 correspondence between principal 7",-extensions and cluster-generated nearnesses. This specializes to a 1-1 match between principal Trcompactifications and contigua! nearnesses.These results are utilized to obtain a 1-1 correspondence between Lodato proximities and a subclass of T,-compactifications. Each proximity has a largest compatible nearness, which is contigua! The extension induced by this nearness is the construction of Gagrat and Naimpally and is characterized by the property that the dual of each clan converges. Hence we obtain a 1-1 match between Lodato proximities and clan-complete principal r,-compactifications. When restricted to ¿/-proximities, this correspondence yields the usual map between r2-compactifications and ¿/'-proximities.

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Eingegangen a m 1. 3. 1966) Intr odue tio n In 1936, 31. H. STOKE [4] proved that every topological space can be iiiade into a To-space by identifying topologically indistinguishable points. The topology of the resulting space is lattice isomorphic t o that of the original space. An investigation into the effect of STOXE'S construction on the uniformities (if any) which induce the topology of the original space revealed that there is a nice 1 -1 correspondence between the uniformities compatible with the original topology and those compatible with the new topology. This led us t o inqdire into the possibility of extending to general uniform spaces theorems which are usually proved only for separated uniform spaces. I n particular, we wished to determine t o what extent the properties of a given uniform space are dependent on the points of the space. If we begin with the lattice of open sets of a topological space we can define a uniformity on the lattice in close analogy with the definition by coveIs of a uniformity on the space. (This definition is largely due to TUKEY [5], and can be found in ISBELL'S book [a], p. 4; however, whatwe mean by a uniformity is what ISBELL calls a preuniformity.) The definition by entourages of a uniformity could perhaps also be carried over to a lattice. but would be awkward. since it requires working with the product of the lattice with itself. I n Section 2 it is proved that there is an ordei-preserviiig 1 -1 correspondence between the set of uniformities on a lattice and those 011 a representation of the lattice as a topohgical space. (Section 1 gives a characterization -due to KOKALSKY [3] -of those lattices which are the open sets for some topological space.) T t is also proved in Section 2 that the concepts of pseudometrizability, proximity class, and m-boundedness have their analogies in a uniform lattice. Section 3 deals with completeness (in the sense of CAUCHY filters converging). This idea can also be carried over t o a uniform lattice. where the concept of a funnel (evidently dne t o I 0 Math. Nadir. 19Ob, Bd. 36, H. 5/6

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