Uniform Structures on the Lattices of Open Sets

Abstract

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

Identification spaces and unique uniformity

“…Note that when as and y are distinct points in S, then x ^ y iff x } y eX and x ~ y in X. Thus ^ determines the members of ϊ 7 , with the identification of {t} with £ whenever ίeT\7. It is easy to show that the quotient topology on T agrees with c^f and that (X, ^) is a dense subspace of (S, SS).…”

“…Therefore E = g~\g(E)) and g{E) e JT Hence JTc gf. Then there is ITeJ3T such that It is noted that Theorem 3.5 can also be proved from Theorem 2.1 in [7]. 4* Unique compatible uniformity and proximity* Early in the study of uniform spaces it was observed that a compact, completely regular topological space admits exactly one compatible uniformity [8,Theorem 20.38].…”