CES 6L L f '. .‘, I" mm m: Pa. 2}. n. ~31»? «E 6.8 u... .9 NM“. n)? .5 r: “S :43: .g, 3 ran ;r ' ' K. 3 [3; I 3.3 .( up; "k'l' I: \1' G CGY m m Q gr @5- F! i fv’iiCHE (u '1 '3 L: “C" ,I'tesis 506 Hm: (.1 U. E;3::__::._::_:__:::13;:2. LIBRARY - Michigan State University This is to certilg that the thesis entitled Ir'fien presented bg has been accepted towards fulfillment of the requirements for . AL',“~- 4:4 i" ‘ degree. 111 q ' 7‘ "*3 arm 71% Majur profe@r hate "3" 7” m“. PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before ode we. DATE DUE DATE DUE DATE DUE SU te An Affirmative Action/Emu! Opportmlty lnetltwen own-3 TOPOLOGY IN LATTICES By EDWARD STAFFORD NORTHAM A THESIS Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1953 .. 9’ 9'5 £1 The author wishes to express his sincere appreciation to Professor L. M. Kelly for much helpful advice given throughout the investigation. 363569 Introduction Various intrinsic topologies which can be introduced into a lattice have been defined by Kantorovitdh (5), Birkhoff (2) and Frink (h), and it is with these that the present investigation is concerned. The thesis is divided into three chapters. The first consists, for the most part, of basic definitions and well known theorems concern- ing lattice theory, general topology and intrinsic tepologies in lattices, while the second and third chapters contain an exposition of the results of this ’thesis. These results are summarized on pages 16 and 17 after adequate terminology has been introduced. Mainly they deal with the Hausdorff charadter of the intrinsic topologies. The various numbered problems referred to are from a list which has been compiled by Garrett Birkhoff and appears in (1). This book also contains proofs of the various assertions about lattice theory which are made in Chapter I. It is the standard reference for the subject. The numbers in parentheses refer to the bibliography at the end. An expression of the type {II P(x)} , where x is an elanent of some given set E and P(x) is a property of x, stands far the subset of E consisting of all elements having property P. Chapter I A. B. C. D. Chapter II Chapter III TABLE OF CONTENTS page Preliminary Results eeeeeeeeeeeee l Partly Ordered Sets and Lattices. l FilterBOOOOOOOOOOOOOO00.000000... 7 General Topology................. lO Topology in Lattices............. 1h The Interval TOPOlogyeeeeeeeeeeeee 18 The Order Topology................ 25 BibliographyOOOOOOOOOOOOOOOOOOOOOOOOOOOOO0.... 57 Chapter I Preliminary Results A. Partly Ordered Sets and Lattices A partly ordered set is a collection of elements and a binary relation defined on‘ these elements which is reflexive, asymmetric and transitive. If we denote the relation by 5 then the three axioms for a partly ordered set are: (i) x _<_ 1 (ii) 15 y and y5 1 imply x=y (iii) 15y andy5z implyx5z. We may write y _>_ 1: instead of x 5 y and in this case we may sayyisgzggxorxisgggggy. Ifx5yandxfy'we write x < y and say that x is properly £13335 y. Given a partly ordered set P we can construct another 1”, called the dual of P, bys aying that 1 5y in P' if and only ify511n P. Ifx5y, x=y ory5x, we saythatxand y are comparable. Otherwise they are called incongaarable. Most partly ordered sets which have mathematical significance satisfy certain other axioms. The least restrictive of the axioms we shall consider is the com- positive axiom of E. H. Moore. A partly ordered set is compositive or directed, as we say nowadays, if given 1 and y there exists 2 such that x 5 z and y 5 2. If a directed set D is mapped into another set S by a mapping f, we call the resulting pair (D,f) a directed system taking values in S. A directed systan is thus a gen- p.2 eralized sequence, and in fact the use of diredted systems is essential if the usual convergence statements concerning sequences on the real line are to be carried over to fairly general topological spaces. A terminal subset T of a directed setais a collection of the type {1| 1 Z a} for some a 6 D. If {y} is any collection of elements of a partly ordered set P we say that u is an pppg; pgu_r_1_d of {at} if a, 5ufor all a, . Anelementuisalpggppppeppgufl (or lub) of {at} if u is an upper bound and is under any other upper bound. A lub is clearly unique. In a similar manner we define lpggg; pgpgd and greatest 2.9.15.1: 19311:; (glb). If a partly ordered set is such that any pair of elements has a glb' and a lub it is called a lattice. In this case the glb and lub of x and y are denoted respectively by x n y and x u y . Clearly the Operations n and U are idempotent, commutative, associative and satisfy the absorption laws: 1 u (x n y)==x and x n (x u y)=x. Also xgy if and only ifxu y=x or an y=y. Acne to one mapping of one lattice onto another is called an isomor- phic}. if it preserves lub's and glb's. A lattice is called complete if any collection {at} of elements has a glb and a lub. These are denoted respectively by Aa. and Vat . In particular a complete lattice has a least element, usually denoted by O, and a greatest element, usually denoted by I. In a lattice with O, x and y are p-3 called disjoint if x n y = O. A lattice for which every set having an upper bound has lub and similarly for lower bounds is called conditionally gemplete. The real line under the usual ordering is conditionally complete but not complete. Any conditionally complete lattice can be made complete by adjoining an O or an I or both. Just as on the real line, we can define cpen and closed intervals in a partly ordered set. The _o_p_e_;_1__ interval (x,y) where x < y is {2 l x < s < y) and the closed interval [x,y] is {2 l x 5 z 5 y} . Also we can define semi-infinite intervals in the usual way. [1,03] is {z I z __>_ x} and [-0 ,x] is {2| 2 5 x} . In a lattice we can show that the intersection of two closed intervals is a closed interval. In fact if x 5 [a,b] and x E [c,d] , then a5x5b andc5x5d.HenceaUc5x5bnd or x e [a u c,b n (B and conversely this interval is contained in each of the first two. It should be noted that in general the intersection of two open intervals is not an open interval and that in a partly ordered set the intersection of two closed intervals need not be a closed interval. In the future, unless otherwise stated, all intervals will be assumed to be closed. A lattice having the property that any two elements are comparable is a 93513; and is referred to as being simply ordered. Any subset of a partly ordered set which is a chain in the induced ordering is called a chain of Pour the partly ordered set. An infinite chain not isomorphic to _the positive integers is called a transfinite squence. To see that a lattice is conditionally complete it is not nec- .essary to test every collection of elements for lub and glb. Ronnie has shown (6; p 587) that a lattice is conditionally complete if any chain having lower bound has a glb. An gpgg in a lattice having an O is an element x such that O < y'5 x implies y=x. In this case we say that x covers 0. In general if s < y.5 x implies y=x, we say that x covers 2. Any lattice which satisfies the distributive laws: xu (yn z)=(xu y)n (xv z) and xn (yu z)=(xn y)u (xnz) is called distributive. It should be noted that either of these laws implies the other. Any collection of sets satis- fies these laws and in fact it can be shown that this exam? ple includes all distributive lattices in the sense of isomorphism. A complete or conditionally complete lattice may satisfy either or both of the infinite distributive laws: (i) so (A b‘)=/\(aubd) (11) .n(vb.)=V(.nb,). Each of these implies both of the finite distributive laws but a distributive, lattice may satisfy either, none or both of the infinite distributive laws. In a non-conditionally complete lattice (i) and (11) will be said to hold if the required upper and lower bounds exist and are equal. If (i) or (ii) holds, so does its extension to any finite number of terms: p-S (1)'(AA‘s)U(ABbp)=AAB(Ia U be) .(111'tha. )n (VBb. )=Vu(a.n b. ). Here (at, B ) ranges over the Cartesian product set (A,B). Each of these is a special case of the corresponding doubly infinite distributive law but does not imply it. (“H/cling um] =AFlVC unwifl (11)"AC[V A: um] =VF[A C Huang Each A; ={0l ...} is the index set of a collection of elements of the lattice and B‘ranges over a set C. F is the class of all single valued functions 1, assigning to each 3‘ £0 a value?“ )e Ar, When C has two members, (1)” and (11)" reduce to (i) and (ii) respectively. Complete chains and the lattice of all subsets of any given set are examples which satisfy (1)" and (11)”. Such lattices are called completely distributive. In a lattice with O and I, an element x' is called a complement of x if xnx'=0 and xu 1'=I and a lattice in which every element has a complement is called compl- emented. A complemented distributive lattice is a Boolean algebra. In a Boolean algebra the complements are unique and are orthocomplements ie (a')'=a. Any Boolean algebra is isomorphic to a Boolean algebra of subsets where finite glb and lub and complementation have their usual set-theoretic interpretation. p.6 Next we shall define several methods for building new partly ordered sets out of given ones. The simplest process is ordinal addition. If L and M are (non-over- lapping) partly ordered sets, then their ordinal sum L am is constructed out of the sum of the two sets by putting the elements of L and M in their given order and then putting every element of L under every element of M. In general this sum is non-commutative and always the ordinal sum of two chains is a chain. This definition can be extended to an arbitrary number on non-overlapping sum- mands. If M={m, m', m"...} is a lattice and {Lmlis a collection of lattices indexed by it we can define:M I'm as follows: l'm,5 I'm. if and only if m' < m" or m'=m" and l'n,5 l'm. in I‘m" Later we will be interested in the case where L and M are chains and in this case}:M In is clearly a chain too. The ordinal product L0H is defined on the Cartesean product set and here we say (l',m')5(l',m") if and only if l' mxn= x=V A x . It is easily seen that on the real line m n _>_ m n this is equivalent to ordinary convergence. A set x is p-15 called closed if xnex and xnfax imply'xsx . We could Just as easily have defined order convergence to x of an arbitrary directed set xuby requiring that \A*meg=xe \/8 Aasz“ and then we would say that a set x is closed if xas X and x, _.,x imply xe X. Both of these definitions can be extended immediately to lattices. The statement x.e-§.x carries with it the tacit assertion that the various infinite upper and lower bounds in the definition exist. The tOpology using ordinary sequences was first investigated by Kantorovitch (5) in the special case of Abelian l-groups. He discussed applications to conver- gence in measure and functions of bounded variation. Sbmetimes this tOpology is named after him. The gen- eralization to directed sets is due to Garrett Birkhoff (2) and the tOpology so obtained is called.the 2392p tepology of a lattice. Clearly the Kantorovitch topology is finer than the order t0pology. If we use transfinite sequences in the above we get a third type of convergence topology, finer than the order topology but coarser than the Kantorovitch topology, which we call the sequential pgggg topology. Sbme of its properties have been det- ermined by Ronnie in (6) and (7). As for the tepologies based on open and closed in- tervals, the former is discrete on the cardinal product ((x,y)5(x',y') if x5x' and y5y') of the real line with itself and apparently it has not been considered worth investigating. The topology using closed intervals p.16 was introduced by Frink (h) and is known as the interval topology. This tepolcgy has the rather interesting property of being compact on complete lattices and further- more it agrees with the order tepology on the product of a closed line segment with itself and in other cases. It is easy to see that a closed interval of a lattice is closed in the order topology, which means that the order tepology is always finer than the interval topology. we see at once that all of the topologies defined so far are always T1. We are now in a position to describe the results of the present investigation. Birkhoff's Problem 76 asks if the order and interval t0pologies agree in a complete Boolean algebra. This is answered in the neg- ative by an example of a Boolean algebra which is Hausdorff in its order topology but not in the interval tOpology. This problem has already been solved by Rennie (7). Our approach first of all leads to a necessary condition (Theorem 2.2) that the interval tepology be Hausdorff and this condition gives an easy (negative) answer to Problem 10h: I'Is any l-group a topological group and a topolog- ical lattice in its interval tepology ?' Also in Theorem.2.5 we find a.necessary and sufficient condition that the interval tepology of a Boolean algebra be Hausdorff. These ideas further lead to a solution to part of Problem 21. We find a necessary and sufficient condition that an element of a lattice be isolated in the interval p.17 topology (Theorem 2.7). This completes the main results of Chapter II. Chapter III deals, for the most part with the Hausdorff character of the order topology. Theorem 15 of Birkhoff (l; p 60) asserts that the order topology is always Hausdorff, but Professor Birkhoff has recently agreed that the proof given is inadequate and suggests that the problem is an interesting open question. By the introd- uction of order convergent filters we are led to a lattice in which the sequential order topology is not Hausdorff. Since this lattice is complete, the In.t°P01°8y 0f Rennie is non-Hausdorff too. This answers Problem 2. of Rennie (6). We are abld to show that at least in a complete, completely distributive lattice the order t0pology is always Hausdorff. Finally as a side result we answer the other part of Birkhoff's Problem 21, which is to find a necessary and sufficient condition that an element be isolated in the order tepology. pal-8 Chapter II The Interval Topology First we shall investigate the Hausdorff character of the interval t0pology in a general partly ordered set. It follows at once from the definition of a basis for open sets that if x and y are any two points in a Hausdorff space and B is any basis for the cpen sets, then x and y can be separated by open sets from B, say, U and V. Looking at the complements of U and V we Obtain the dual require- ment that given any two distinct points, the space can be covered by two closed sets each of which contains exactly one of the points, and in addition we may select these sets from any given basis for the closed sets. In particular: Theorem 2.1. The interval tepology‘pf'g partly ordered set 23 Hausdorff £2 and only if given any two distinct points 311353 lg _a_ covering fling £93 ‘91 BEES: ofug finite number pf closed intervals such‘ppg£_ no interval contains pppp point . To obtain a necessary condition that the interval topology of a lattice be Hausdorff we look at any pair of comparable elements, x < y, for Which by Theorem 2.1 there is a covering of the lattice by a finite number of closed intervals such that no interval contains [x,y]. Taking the trace (intersection) on [x,y] of each member of the covering we obtain a covering of [x,y] by a finite number of closed subintervals, no one of which is [x,y] itself. In other words, if we exclude x and y, each point offix, i] is comparable with at least one of the p.19 remaining and points of the subintervals. The same is true if either x or y is infinite. Let us say that a collection of elements {a1} is a separating set of the interval [x,y] if x < a1 < y for each a1 and every element of [x,y'] is comparable with at least one of the a1. If y covers x we will agree that the empty set separates [x,y] . Summarizing we have: Theorem 2.2. g necessary condition for the interval togology 2f a lattice 1:2 333 Hausdorff _ig that every closed interval have _a finite separating set {res}. We are now in a position to prove: Theorem 2g. In _a_ Boolean algebra without atoms, the interval 0,1 has no fss. Proof. If {a1, a ... an} is a fee, adjoin the 2: complements ai of the a1 obtaining a new set B. For each subset of B form the meet of its elements andfrom this collection of meets let {or c2, ... ck} be the non null minimal ones. It is convenient to think in terms of sets in which case the c1 are a collection of disjoints sets whose union intersects each ak and its complement. Now for each‘ci choose d1 so that 04d1 < c1 and let d=d uda ...ucik . Then since a > dna we have dial 1 i i and since dnai > 0, d ,4 a1‘ . In other words d is not comparable with any a1. Remark. The lattice of all measurable subsets (modulo sets of measure zero) of the unit interval is a complete Boolean algebra without atoms, so its interval t0pology p.20 is, by the preceeding theorem, not Hausdorff whereas the order t0pology is (l; p 169 and p 80). Theorem 2.5 may be applied to the solution of problem 76 of (1). This problem has already been solved by B.C. Rennie (7) using a different method. We might observe further, that an examination of the prnof of Theorem 2.5 shows that the following someWhat more general result may be established. Theorem 2.h_.'§ distributive lattice without atoms, 22 which each element (except I) has 2 non null disjoint element,.i§ not Hausdorff Ea its interval topology. Theorem 2:5 . The interval topology gf‘g Boolean algebra i5 Hausdorff if and only if every element :3 over £9. 2.322. Proof. If some element x is over no atom, then the interval[0,i] is a Boolean algebra without atoms. Hence, by Theorem.2.5, it has no fss and thus from.Theorem 2.2 the t0pology is not Hausdorff. Assume then that every element is over an atom and let x and y be any pair of distinct elements. Since xr\y' and yrxx' cannot both be null there must be an atom a under, say, 1 but not y. It follows at once that the intervals [a,I] and [0,af] are disjoint closed intervals which cover the algebra, and the t0pology is Hausdorff (Theorem 2.1L Next we apply Theorem 2.2 to Problem 10h of (l), which should read: '18 any l-group a topological group and a tepolcgical lattice in its interval t0pology 2' p.21 Since the interval t0p010gy is T1 it must be Hausdorff if the 1-group is to be a tOpological group. Now the additive group of all continuous real valued functions defined on the closed unit interval is an 1-group using the natural ordering (l; p 216). If fo denotes the function f(x)=0 and f denotes the function f(x)=1, we show that 1 the interval [f0, f1] has no fss. If {a1 ... an} were such a set, choose for each ai some point x1 fliers a1(x1)<1. Define a continuous function a(x) to be l-at each of the x1 and elsewhere to take on values between 0 and 1 so that its integral over the interval is less than that of any a1. Clearly a(x)is not comparable with any of the a1. It is interesting‘to note that the set of all real-valued functions does have a fss for any interval and in fact (2) the interval tepology is Hausdorff. We have shown: Theorem 2.6.Iég 1-group need n22.23‘g topOlogical groug‘gg its interval togology. Finally we find a necessary and.sufficient condition for a point x to be isolated in the interval topology of a lattice L. This is part of Problem 21 of (1). First suppose that 0 < x < 1. If x is isolated then er is a closed set and in fact must be the union of a finite number of closed intervals I1 ... Ik . Let P denote the set of elements of L under x and take the trace of each Ik on P, which is a closed interval. From the set of upper endpoints of the traces select the maximal ones. These form a non-empty finite set {x1 ... xn} , each x1 is p.22 covered by x. and any element under x is under some x1. The same argument can be applied to the set of elements over x. Looking at the lower endpoint of each Ik let us replace it by amtx) if it is under, (over) x. Then if an upper endpoint is under (over) x replace it by xt”). Having done this we have a covering of L'by a finite number of closed intervals for which none of the endpoints (except possibly x, O, or I) is comparable with x. In 'other words x belongs to a fss of L in.which no other member is comparable with x, and we have shown the necessity of the conditions in the following Theorem 2.1: The following conditions are necessary interval tepology‘gg‘g lattice L. (a) x covers 3 finite number 2; elements and every covers Xe (c) 1: belongs £3 3 fss if. L in which pg other member 1.3 comparable with 1. It is ealy to see that the above conditions are sufficient. If the fss is {x,a1 ... ak} and if x covers {b1 ... bm} and if x is covered by {cl ... on} then.L~x is the union of the following intervals: ['“5‘13 [81,00] [-m'bi] [01,00] for all permissible values of i. If x is O or I then clearly (b) or (a) is necessary and sufficient for x to be isolated. p-25 We shall conclude our discussion of the interval tepolOgy with an example which shows that the necessary condition of Theorem 2.2 is not sufficient and that cond- ition 5 of Theorem 2.7 is not a consequence of the first two 0 o The lattice is formed by all finite and infinite sequences (an) which take values in a two element set, say {x,y}, and a t0p element I. We say that (an) 5 (tn ) if (tn) is a continuation of (an). The diagram has been arranged so that at any given term of a sequence an x means ”take the left branch" and a y means ” take the right branch”. Thus the circled point stands for the finite sequence (xyx). Let us call the points corresponding to finite sequences,finite points, and those corresponding to infinite sequences, infinite points. It is obvious that there are an uncountable number of infinite points over any finite point, hence any finite collection of intervals whose upper endppint is I contains all infinite points p-P-h or does not contain an uncountable number of them. Any other type of interval can contain only a countable number of elements. In other words if the lattice is covered by a finite number of intervals in any way, each infinite point is contained in an interval whose upper end point is I. By Theorem.2.l this means that no infinite point can be separated from I by open sets, so the interval topology of this lattice is not Hausdorff. If we insert an element u between some infinite point 2, say (1, x, ...), and I, we get a lattice in which u satisfies the first two conditions of Theorem.2.7 but not the third. Consider the elements not on the chain [0,2] but which cover members of it. There are an infinite number of these,{2n}, none of which are com- parable with 2. Furthermore the intervals [2n,I] are disjoint except for I (which cannot belong to a fss), and any two intervals [mag and [c,zm] have intersection contained in the chain [0,2] . This means that the 2n are comparable with no finite collection of points, none of which is on the chain [0,2] . Thus any fss of the lattice must contain some point of [0,2] , hence a point comparable with u. p.25 Chapter III The Order Topology When solving Birkhoff's Problem 76 (1; p 166) we might have used our Theorem 2.5 together with Theorem 13 of Birkhoff (1; p 60 ) to get a large class of Boolean algebras for which the order and interval topologies do not agree. This theorem asserts that the order topology of any partly ordered set is Hausdorff. The argument given there proceeds as follows. First it is noted that if a directed set order converges, then the limit is unique. This follows immediately from the uniqueness of glb and lub. Secondly reference is made to the fact that in any tOpological space, if directed systems have unique limits, then the space is Hausdorff. Now in the order topology, in general, there are convergent directed systems which do not order converge and we must show that these have unique limits too, if we are to argue in this manner. It turns out however that the order topology can have convergent directed systems with non-unique limits, and hence need not be Hausdorff. The main difficulty when working with the order topology is that one must take into consideration not only a large variety of directed sets but also the various ways each can be mapped into the lattice. Tukey (9) has shown that actually we need consider only very special types of directed.sets, but nevertheless the direct attack on the Hausdorff character seems p.26 difficult to carry through. By the introduction of order convergent filters and use of the equivalence between directed sets and filters we shall reduce the problem to a discussion of the intervals on the lattice, and describe a process whereby the Hausdorff character may be determined in certain cases. Throughout we shall denote filters, and collections of sets in general, by script letters. Individual sets of a collection will be denoted by capital Roman letters and as usual small Roman letters will stand for elements of a lattice. By definition, a filter base 8.={B,} order ccnvergps to x if AJ¢=x= Vm. where B¢=fx:}, ja=Vx:, and mat-'Ax: . Since any two sets of a filter base'have non empty intersection it is clear that every up is under every 1,. This implies that if a filter base order converges to x then so does any finer filter base. The following lemma is an immediate consequence of the definitions. Lemma 5.1 A directed system order converges tg'x ‘if and only if the associated filter base order converges _t_9_ x. mg _a_ filter p_s_1_s__e_ 933g converges _1_;_o_ _x .i_f_ gig _o_n_l_y [if the associated directed gystem order convepges 32 x. New for each memeber of a filter base 63.: {Bl} order converging to x, we define m6 and J“ as above and then assign to Ba the interval [111.” L] , which contains B... Each interval contains x and the collection of intervals forms a system of generators for a new filter which clearly order converges to x too. If B... , 8,6 B then Bdn BpC [m,, 1,3 0 [m,, 1,] nL‘m‘u mfi, Ln 3,] , which tells p-27 us that the generated filter base is actually coarser than B. Summarizing we have Lemma .2 Evepy filter base order converginggtg_ x .is.finar.than.s filter basa.af.interxals, which.1lkewlas order converges tgfix. Let us call the intersection of all filters order converging to x the filter of pseudo-neighborhoods of x‘. Since every filter base order converging to x must converge tox in the order t0pology, it follows that the filter of pseudo-nbhds must converge in the order topology. If the filter of pseudo-nbhds satisfied the nbhd axioms, it would indeed be the nbhd filter, but there are cases where the nbhd filter is properly coarser than the pseudo-nbhd filter. Using Lemma 5.2 we see that in order to obtain the pseudo-nbhd filter of x we need only consider the intersection of the filter bases of intervals which order converge to x, and it is easy to decide whether a filter base of intervals order converges t o it . Lemma §;3_ g.filter base 9;,intervals ggdgr oonverggs §p_x i£.and only_i__§h9_intersegtion_g£,gll tha.intazlals.is.x. A basis for the pseudo-nbhd filter is formed than as follows. From each collection of intervals having the finite intersection prOperty (the intersection of any p.28 two intervals of the collection is a member of the collection) whose intersection is x, select an interval and form the union of the selected intervals. The totality of all such unions is a basis for the pseudo-nbhd filter. ' Now in order that the order topology of a lattice be Hausdorff it is necessary (but probably not sufficient) that for any two distinct points x and y, their pseudo- nbhd filters be disjoint. The next paragraphs will be devoted to the construction of a lattice where this necessary condition is not satisfied, but first we note Lemma 1.1; The filterugg nbhds if x _ig disjoint from any filter basefigf intervals which order converges £2.y- Proof: We can pick any interval G ofib which does not contain x. Then if every set of”u had non empty trace on G these traces would form a filter base on G which would converge in the order topology to x, but this is impossible since 0 is closed in the order topology. In a lattice, to show that every pseudo-nbhd of x intersects every pseudo-nbhd of y, it is necessary and sufficient to find a collection of filter bases of intervals, each order converging to x, such that no matter how we select an interval from.aaoh base and form.the union, some filter base order converging to y has non empty trace on this union. The necessity is ob- vious. p.29 The sufficiency follows from the fact that each pseudo- nbhd contains one of the above unions. If the filter bases are nested, then the (transfinite) sequential order topology will be non Hausdorff. Let us denote the first uncountable ordinal number byid and let A be the 1 dual ofLJl +-l. A is a complete chain and for the sake of convenience we denote its first element by O. The set of all ordinary sequences taking values in A with the natural (componentwise) ordering is a complete lattice and we obtain a lattice with non Hausdorff sequential order topology if we restrict ourself to those sequences for which all but a finite number of values are 0 and then adjoin a top element I. This lattice L is clearly condition- ally complete and since it has a top element is complete. Now we shall exhibit a collection of nested filter bases of intervals, each order converging to the sequence (0,0 ...), which we shall henceforth denote by 0'. The bases are of the type {[(a,o,o,...),o']} {[(ma,0,o,...),oj} , {flr,r,r,o,o,...),o_']} etc. Where{a,p,y,...} range over A and are not 0. If we pick one interval from each basis we shall have selected, as upper end points, a countable collection of elements of A. There is an element of A which is under each upper end point and yet properly over 0. Now each member of the following sequence of elements of L is in the union of the selected intervals p.30 (A,o,o,...). (A,A,o,o,...), (z,),7.,o,o,...)... and the sequence order converges to I. The associated filter base likewise order converges to I and has non empty trace on the union, hence the sequential order topology is not Hausdorff. Theorem 5:5 The sequential order topology gf‘g lattice pggg‘ggt_pg Hausdorff. This example also provides an answertzo a problem posed by Rennie (6; p too) as to whether the L-topology of a lattice is always Hausdorff. He defines the L-t0pology of a lattice by taking as a basis for the Open sets those sets which intersect each maximal chain in an open interval (of the chain) and are convex. That is if a and.b belong to a set, a < b, then all c such that a §,c'5 b belong to the set. A maximal chain is one which is contained in no other chain. Rennie has shown (7; p 20) that in a complete lattice the Lntopology is coarser than the(transfinite)sequential order topology. Thus we have the Corollaryg3.6 The L-topology'gfflg lattice need not 1E2 Hausdorff. The following somewhat more direct argument shows that the sequential order tOpology of the above example is not Hausdorff. If U is any nbhd of 0 we see by the equivalence of directed sets and filters that it must contain some interyal out of every nested collection whose intersection is 0. Hence U contains a sequence p.51 converging to I and any nbhd of I must intersect U. The more roundabout argument is presented because the concept of pseudo-nbhd enables us to describe a process whereby the Hausdorff property may be established in certain cases. It is easy to verify the following lemma which relates pseudo-nbhds to cpen sets. Lemma 5.] A set i open i3 the order topology if This means that any open set containing an element x can be constructed in the following manner. First take a pseudo-nbhd of x. Then choose a pseudo-nEUd of each of its points and form the union E whith will be called a pZ-nbhd of x. Having defined a pn-nbhd V of x, we define a pngfI-nbhd by selecting for each point of V a pseudo- nbhd and taking the union. The union of all the pn-nbhds is clearly an open set. Now we shall apply the above process to a complete, completely distributive lattice, where we have m-\/,,.[/\...u.,..] =/\F[vc um] and (11)'/\c [Viv “33“] =VF [Ac ”3344”) as stated on page 5. If C indexes the set of filter bases of intervals order converging x and the u” are the upper (lower) endpoints of the filter base A; in (i)'((ii)9) then the left sides of (1)” and (ii)“ are p.32 x and we see that the pseudo-nbhds order converge to x and have a basis of intervals. Since order convergent filters have unique limits, this means that the pseudo-nbhd filters are disjoint. So given any distinct elements x and y we have disjoint pseudo-nbhds [a1,b1],[c1,dl] of x and y respectively. Focusing attention on the former, we see that if we select a pseudo-nbhd [uuvtj for each point of [a1,b1] and form the union of these sets, this union is contained in the interval [Ant , VvJ , which is a pz-nbhd of x containing C‘ybfl . It follows from the infinite distributive laws that the intersection all such p2-nbhds is exactly [‘l'blj . If each of these pZ-nbhds had non empty trace on [c1 ,d1] , these traces would form a filter base of intervals, for which the sets would have non empty intersection since every upper end point must be over every lower end point. This is a contradiction so there must be some interval [a2,b2] which is a pZ-nbhd of x and is disjoint from [31,le . Then we apply the same argument to [c1,d1] and get a pz-nbhd [c2,d2] of y disjoint from [a2,b2]. Continuing in this manner we obtain ascending sequences of intervals whose unions will be disjoint Open sets containing x and y respectively. Thus we have established Theorem 1.8 TE order topology pf _a_ complete P53 completely distributive lattice is Hausdorff. The first part of Birkhoff's PrOblem 21 (l; p 62) is concerned with finding necessary and sufficient conditions that an element x be isolated in the order topology. By Lemma 5.7 this will be the case if and only if x is a pseudo-nbhd of itself. In other words [x,x1 belongs to any collection of intervals having finite intersection property whose intersection is x. This shows the necessity of the conditions in the following Theorem 5.9 _Ir_1_ order that x pp 119131251 _ip _tpg order topology pf‘g lattice ;§_;§ negessgry and_ sufficient that any collegtiop of,elements ghgse_ glb ing have 2 finite subset whose glb ig’x.and dually that any collection pf elements whose lub ig x hgyg g finite subset ghppg.glp.ip_x. Proof: To show sufficiency let {Eat ,bg} be a collection of intervals whose intersection is x and suppose that M and N are finite subsets such that Auxm=x= Van. Then the intervals whose upper endpoints are the xm together with.those whose lower endpoints are the x.n form a finite subccllection whose intersection. is x. If the lattice is complete we can use a result found in Viadyanathaswamy (10; p 59) to get much more tractible conditions for x to be isolated. An element x is called a jpmp.element of a lattice if it is not p-Bh the lub of any chain whose members are properly under x and dually if it is not the glb of and chain whose elements are prOperly over x. Theorem 5.10 .IEME complete lattice thprollowing properties p£_pp element x are equivalent: (i) x is isolated in the order tOpology (ii) x is isolated in the sequential order topology (iii) x is 3. jump element (iv) any collection pf elements whose glb ig’x has a finite subset Whose glh is x and any Proof: (1) implies (ii) since the second topology is finer than the first. (ii) implies (iii) since any chain whose glb or lub is x gives rise to a transfinite sequence order converging to x. The above cited result of Viadyanathaswamy is essentially the statement that (iii) implies (iv) and it follows from.Theorem 5.9 that (iv) implies (1). One problem in this area, as yet unsolved, is to decide Whether the Kantorovitch topology is always Hausdorff. This tOpology is finer than the others and can be Hausdorff, for instance in the example¢of Theorem 5.5, where the others are not. Theorem.72.5 of Vaidyanathaswamy (10; p 275) asserts that any p-55 sequential convergence scheme with unique limits gives rise to a.Hausdorff topology and would settle this question if it were correct. However there exist sequential convergence schemes having unique limits which do not give rise to Hausdorff topologies. By a sequential convergence scheme having unique limits we mean any process for assigning limits to sequences (if x is seeigned to x.n we write xnfex) such that (i) if xh=x for all n.then.xnf)x (ii) if xn=yn for all but a finite number of n thenx.n and yn have the same limit or do not converge (iii)if xn-ex and xnj is a subsequence then xnj-Qx (iv) if xn-)x and xn—sy then x=y The derived topology is obtained by calling a set X closed if {xgex and xn-+x imply xe X. Now let us re-define convergence on the closed unit interval of the real line. As required we say'that thexiif xn=x for all but a.finite number of n. If x.n is monotone non-decreasing and does not converge by the previous requirement we say xd—wl, and if xn is monotone non-increasing and does not converge by the first re- quirement we say xfi—90. It is easy to see that any Open set containing 0 but not 1 must have its complement well ordered and hence countable. Similarly any cpen set containing 1 but not 0 must have bountable complement p.56 so 0 and 1 cannot be separated by open sets. In conclusion we indicate various extensions of the results obtained here which might be expected. First of all there is a large gap between the example in Theorem 5.5 and the complete, completely distributive lattices, so far the only extensive class for which we have been able to verify the Hausdorff character. The lattice of Theorem 5.5 is distributive and in fact satisfies the infinite distributive law (1) but not (ii). Among the conditionally complete lattices satisfying (1) and (ii) we find the (conditionally complete) Boolean algebras and l-groups, both of which are of interest in certain applications. Theorem 18 of Birkhoff (1; p 251) implies the Hausdorff character in the latter case, but the proof seems open to the same objections discussed here on page 25. In view of the essentially negative results obtained, it seems reasonable that future studies of lattice tOpologies will be fruitful only if restricted to the lattices which enter in the applications. So far none of the lattice topolOgies has given much insight into the structure of lattices in general. 9-57 BIBLIOGRAPHY (l) G. Birkhoff, Lattice Theory, Am. Math. Soc. 0011. Pub. Vol 25 (Rev. ed.) New York, 19u8. (2) G.Birkhoff, Moore Smith Convergence in General TOpology, Ann. of Math. 58 (1957) (5) N. Bourbaki, Elements dp Mathematique, Premiere partie, Livre III, Ch. I, Paris, 1951. (h) 0. Frink, Topology ip Lattices, Trans. Am. Math. Soc. 51 (19142) (5) L. Kantorovitch, Lineare Halbgeordnete Raume, Mat. Sbornik 2 (an) (1957). (6) B. C. Rennie, Lattices, Proc. London Math. Soc. Series 2 52 Part 5 (1951). (7) B.C. Rennie,_Thgg£y_pfiggppigg§, Cambridge 1951. (8) J.Schmidt, Beitrage zur Filtertheorie, Math. Nach. (9) J.Tukey, Convergence and Uniformity lg TOpology, Princeton, 19h0. (10) R. Vaidyanathaswamy, Treatise pp Set Tppology, madrafl , 1911.7 0 ‘7 "'TITIi‘flujl'iILflfliliflilfiifljflfl’l’lfliflljfliflflfllflms