Not every Lodato proximity is covered

Abstract

In a recent paper Reed wrote, "In fact it may be that all Lodato proximities are covered. I was unable to find a counterexample". (Remark 1.10)The purpose of this note is to show that, in general, Lodato proximities are not covered. Preliminaries.A closed filter &on a topological space (X, c) is a proper filter (that is, a filter which does not contain the empty set) which has a base consisting of only closed sets. Maximal (with respect to set inclusion) closed filters are all called ultraclosed filters. For more information on the concept of ultraclosed filters see Thron [3].Ultrafilters are maximal proper filters on a set and grills are exactly the unions of ultrafilters. For a detailed discussion on ultrafilters and grills, see Thron [2].A basic proximity πona set X is a symmetric binary relation on the power set &*( X) of X satisfying the conditions:The pair (X, TΓ) is called a basic proximity space provided π is a basic proximity on X.For a basic proximity π on X, we define c v (A)= {χ€l:({jc}^)Gίr} for allΛ C X. A Wallman π-clan is a π-clan which contains some ultraclosed filter. The proximity π is said to be covered if for each (A, B) e 77 there exists a Wallman 77-clan ^such that {A, B) c ^. It is easily verified thatWe conclude this section by proving the following results which will be used to make the final conclusion. PROPOSITION. Let °Ube an ultraclosed filter on (X, c) andstfa base of tf/ consisting of closed sets. If F is a closed set and F Π A Φ 0 for all A in s/then FeiProof. Let <% be the collection of all finite intersections of members of the family s/ U {F}. Then 3& is a filter base consisting of closed sets. Let 0 be the filter generated by 3S as a base. Then ^0 is a closed filter and % D <2f U {F}. By the maximality of <%'\i follows that F e i Proof. Let °ll be an ultraclosed filter on (X, c).Since the space is compact if follows that there exists an x in X such that x e c(F) for all F G ^. Let F be an open neighbourhood of x. Then KΠi 7^ 0 for all F e ^. Thus by the above corollary, KeΦ. Hence ^converges to JC. PROPOSITION. On a T λ -space (X, c), every convergent ultraclosed filter has the form °il(x), for some x e X, where °lί{x) = {A a X: x ^ A}.Proof. Let °ll be an ultraclosed filter on (X, c) such that it converges to a point x e X. Obviously x ^ c(F) for all F ^ °ll. Hence, in particular,