BINARY CONVEXITIES AND DISTRIBUTIVE LATTICES

BINARY CONVEXITIES AND DISTRIBUTIVE LATTICES M. VAN DE VEL [Received 11 August 1982—Revised 7 February 1983]

ABSTRACT A convex structure is binary if every finite family of pairwise intersecting convex sets has a non-empty intersection. Distributive lattices with the convexity of all order-convex sublattices are a prominent type of example, because they correspond exactly to the intervals of a binary convex structure which has a certain separation property. In one direction, this result relies on a study of so-called base-point orders induced by a convex structure. These orderings are used to construct an 'intrinsic' topology. For binary convexities, certain basic questions are answered with the aid of some results on completely distributive lattices. Several applications are given. Dimension problems are studied in a subsequent paper.

0. Introduction A convex structure consists of a set X, together with a collection ^ of subsets of X, henceforth called convex sets, such that (1.1) the empty set and the set X are convex; (1.2) the intersection of convex sets is convex; (1.3) the union of an updirected collection of convex sets is convex. The collection (€ itself is called a convexity on X. Axiom (1.2) allows the construction of an associated (convex) hull operator (usually denoted by lhJ) in the obvious way. The hull of a finite set is called a polytope, and the hull of a two-point set is also called an interval. A half-space is a convex set with a convex complement. The following separation axioms—comparable with the axioms T l 5 ...,T 4 , in topology—are used frequently: Sx: S2: S3: S4:

singletons are convex (which we will assume throughout); two distinct points are in complementary half-spaces; a convex set is an intersection of half-spaces; two disjoint convex sets extend to complementary half-spaces.

In this paper we will concentrate largely on a particular, though fundamental, class of convexities with the following binarity property: each finite collection of pairwise intersecting convex sets has a non-empty intersection. The basic types of examples are described in §1. Many of these examples arise in a topological context; for other examples, a natural topology can be constructed. Binary convex structures on a topological space have been studied extensively in the past years [15, 17, 19, 22, 29, 34, 38]. The topology and the convexity are always assumed to be compatible in the sense that polytopes are closed. A triple, consisting of a set X, a convexity (€, and a compatible topology ST, is called a topological convex Proc. London Math. Soc. (3), 48 (1984), 1-33.

5388.3.48

A

M. VAN DE VEL

structure. Frequent use is made of the following 'topo-convex' separation property: (X,(tf,&~) is called normal provided that for each pair of disjoint convex closed sets Cl,C2 there exist convex closed sets Dl,D2 with

(note that since singletons are assumed convex, the underlying space of a normal convexity must be T2). Equivalently [22, 2.1] each pair of disjoint convex closed sets can be separated by a continuous convexity-preserving (CP) function / : X -> [0,1]. Here, 'convexity-preserving' means that the inverse image of a convex set is convex; the order-convex sets of [0,1] are taken convex. Note that the sets of type /"'PUD

or

/-'[M]

are half-spaces of X. In particular, disjoint polytopes extend to complementary halfspaces, which by [10,1.9] suffices for {X,^) to be S4. If X is any topological convex structure, then the corresponding weak topology o{X is the one generated by the convex closed sets. It is shown in [33, 2.4] that a binary S4 convex structure admits at most one compatible weak topology for which polytopes are compact and for which the convexity is normal. This suggests the existence of an intrinsically defined topology. Our main purpose is to construct such a topology, and to investigate the following problems: (2.1) When is the intrinsic topology T2? (2.2) When are polytopes closed, or compact, in the intrinsic topology? (2.3) When is the convexity normal in the intrinsic topology? We note that for non-binary convexities on non-compact spaces, normality is a rather unrealistic condition: it was observed in [30] that the ordinary convex sets of a topological vector space of dimension greater than 1 never form a normal convexity. Some natural, weaker separation properties on the 'regularity' level have been introduced in [30]. Our main tools in constructing an intrinsic topology are: a notion of induced basepoint orders, and a related notion of (conditional) completeness. The intrinsic topology is the one generated by the complete convex sets. These tools are already available on any S 3 convex structure, and a few non-binary examples may illustrate the naturality of our construction. (3.1) On a vector space with ordinary convex sets, the intrinsic topology is the weak finite topology. See [13] for results on the finite topology. On W, one obtains the Euclidean topology. (3.2) On a complete continuous semilattice, with the order-convex subsemilattices as convex sets, the intrinsic topology is the so-called Lawson topology (see [5]). Apart from these examples, we have confined ourselves to binary convexities. It is shown that each interval becomes a distributive lattice under a suitable base-point order and, conversely, that every distributive lattice with a top and a bottom element is a binary S 4 interval. The intrinsic topology of a complete and distributive lattice

BINARY CONVEXITIES AND DISTRIBUTIVE LATTICES

3

equals the so-called interval topology, and lattice homomorphisms are CP functions. Our main results are the following: (4.\) If a binary S 4 convex structure is complete in each of its base-point orders, then all poly topes are closed and compact. (4.2) (same assumptions). The intrinsic topology is T2 if and only if each interval is completely distributive, which is so if and only if the convexity is normal in its intrinsic topology. (4.3) The weak topology of a normal binary convex structure with compact poly topes is exactly the intrinsic topology. The result in (4.2) should be compared with what is known for complete and distributive lattices: complete distributivity, Hausdorffness of the interval topology, and the existence of enough characters (continuous lattice homomorphisms into [0,1]), are equivalent properties [27, Theorems 6,7]. In fact, this result is essential in our proof of (4.2). These investigations have also led to a number of by-products, for instance: (5.1) Let X, Y be normal binary convex structures with compact polytopes. Then a surjective CP function f: X -> Y is weakly continuous (that is, continuous in the weak topology of its domain and range) if and only if its fibres are closed. (5.2) Let X, Y be as above, let Z be a compact T 2 space, and let f: Z x X ->• Y be continuous in each variable separately and CP in the second variable. Then f is weakly continuous provided that for each z e Z, the hull of f({z} x X) is dense in Y. This paper is organized as follows. After presenting some examples of binary convexities in § 1, we give a general introduction to the theory of binary S 4 convex structures in § 2. The main concept is a notion of 'nearest-point' function, which has already proved useful in topological circumstances [16, 17, 29]. One result on nonbinary convexities is derived to illustrate some differences with the binary theory. In § 3 we discuss induced base-point orders, and we relate binary S 4 convexities with distributive lattices. One consequence is that binary S 4 structures are actually of an algebraic nature. The intrinsic topology is studied in §4 with the aid of some lattice theory. Finally, we present some related results, applications, and unsolved problems (mainly for non-binary convexities) in § 5. We will continue our study of topological binary convex structures in a subsequent paper [35], where dimension-theoretic questions are in order. General references for abstract convexity theory are [10, 25, 30]. Most of the terminology for convex structures that we will need has already been presented above. A few other concepts are explained below at the appropriate places. For terminology and results concerning lattice theory, the reader may consult [1, 5, 6]. Little knowledge of topology is required, except, perhaps in §5. 1. Binary convexity: examples 1.1. Totally ordered sets. If a set X is equipped with a partial order relation, then a subset A of X is called order-convex provided that if ax ^ x ^ a2 and a1,a2 e A then

4

M. VAN DE VEL

x e A. This leads to a convexity on X which, in the case of a total order, is a particularly simple one: its half-spaces are exactly the left and right segments, and it is easily seen to be a binary S 4 convexity. On a totally ordered set one usually considers the order-topology, which is generated by the closed segments. This leads to a topological convex structure. 1.2. Trees. A tree is a connected and locally connected T 2 space in which every two distinct points can be separated by a third one. According to [39,9.1], the intersection of a family of connected subsets of a tree is again connected. Hence, a tree can be convexified by taking connected sets to be convex. It was shown in [30, 2.10] that this convexity is normal binary and has compact polytopes. If X is a tree and if b e X, then the corresponding cutpoint ordering ^b is obtained as follows: y ^ 6 x if and only if b and x are separated by y, or y = b, or y = x. This gives a partial order on X such that for each x e X, the collection

is totally ordered under ^fc. Also, b is the smallest element of < b . A partial converse to these results is given in § 5 below. There is a remarkable relationship between the topology and the convexity of a tree: according to [18, 1.7], a tree carries only one normal binary convexity with compact polytopes, namely the above one (this was shown in [18] for compact trees only, but the argument works equally well for non-compact ones with the aid of Theorem 2.9 below). On the other hand, if the convexity of a tree is given, then the topology is not uniquely determined. For instance, one can pass to the weak topology (for compact spaces with a normal convexity, this does not change the topology); or one can construct a 'fine' topology as described in 5.9 below. The reader may verify that all resulting spaces are trees again, with the same connected sets. 1.3. Lattices. Let (L, A , v ) be a lattice. Call a subset convex provided it is an order-convex sublattice of L. This gives a convexity on L which was formally introduced by Jamison in [12], and which will also be considered throughout this paper. If x l s ...,x n e L, then the hull of {x l5 ...,*„} is the set

{x: Ax( C

which retracts X onto C. Henceforth, p will be called a nearest-point function. 2.3. LEMMA. Let X be a binary convex structure in which every two points can be screened by convex sets. Let F a X be non-empty and finite. Then (1) h(F) has nearest-points; in fact, if b e X, then its nearest-point in h(F) is the unique point in f]yeFh{b,y] n h(F), (2) x e h(F) if and only if f]yeFh{x,y} = {x}.


9

Proof of (1). The convex sets h{x,y}, with y e F, and h(F), meet two by two, and hence they have a point in common. If u,v, with u # v, are both in the intersection, then consider a screening of u, v by convex sets C,D:

As u € h(F), some y e F must be in C. Similarly, some y' e F must be in D.lf x E C, then y cannot be in h{x,y} a C. If x e D, then u cannot be in h{x,y'} c: D. We conclude that the above intersection consists of only one point, c say. If c' were another point in h{x,c] nh(F), then a screening argument as above would lead to another contradiction.

Proof of (2). If x e h{F), then use (1) and the equality (3)

f]h{x,y}= yeF

f]h{x,y}nh(F). yeF

On the other hand, if the left-hand set of (3) consists of x only, then x e h(F) because this left-hand set must meet h(F). 2.4. SCREENING LEMMA. Let X be a binary convex structure. (1) / / C,D are convex sets and if c e C and d e D are such that C n h{c, d} = {c},

D n h{c, d) = {d},

then every two convex sets C',D' screening c,d, also screen C and D. (2) / / the convex sets C,D admit nearest-points, then for each c0 e C, d0 e D, there exist c G C, d e D as above, and such that c,d e h{co,do}. Pairs of points such as c,d above are used frequently. We call them mutual nearest points in C and D, as we did in [33]. Proof o / ( l ) . We have ceC\D',

d€D'\C, CvD' = X.

If C meets D', then by 2.2(1) the nearest-point c in C of d must be in £)', which is a contradiction. One shows similarly that D does not meet C . Proof of (2). Let d be the nearest-point in D of c 0 , and let c be the nearest-point of d in C. By 2.2(1), d e h{co,do}, whence h{co,d}

^h{co,do}.

Again by 2.2(1), c is in h{co,d}, showing that c, d

E

h{c0, d0},

h{c, d) c h{c0, d).

Now 0 # h{c,d}nDcz h{c0,d}nD = {d}, showing that d is also the nearest-point in D of c.

10

M. VAN DE VEL

2.5. THEOREM. A binary convex structure, in which every two points can be screened with convex sets, is S 4 . Hence, the separation axioms S2,S 3 5 S 4 , are equivalent for binary convex structures. Proof. If distinct points can be screened by convex sets, then by Lemmas 2.3 (1) and 2.4, every two disjoint polytopes can be screened by convex sets. By [30, 2.2.], the convexity must then be S 4 . Let us call a convex structure ternary if its Helly number is at most 3. The following result is of independent interest: 2.6.

PROPOSITION.

A ternary S 3 convex structure is S 4 .

The results in Theorem 2.5 and Proposition 2.6 are not paralleled for higher Helly numbers: in [32] an example is given of a convex structure on R3 with Helly number 4 which is S 3 but not S 4 . The proof is based on an example of Bourgin [3] of a 3-simplex which cannot be separated from a translated copy by a hyperplane parallel to one of its faces. Proof. Let ^ be a convexity on X, and let 34? c X, which we will call the median operator of X. The following is an algebraic characterization of binary S2 convexities. 3.6. COROLLARY. Up to isomorphism, the binary S2 convex structures correspond to the median-stable subsets of distributive lattices, equipped with the relative convexity. By an isomorphism we mean a CP bijection with a CP inverse. Proof. Let X be a binary S2 convex structure. If u and v, with u ^ v, are in X, then there is a half-space H of X with u e H, v $ H. This leads us to a CP function X -> {0,1} (where {0,1} is convexified in the obvious way), sending x to 0 if and only if x e H. Together, these functions determine an injective CP function / : X -> {0,1}* = L from X into a suitable power of {0,1}, equipped with the product convexity (see 1.7). The latter equals the usual convexity of the distributive lattice L. By Theorem 2.9(4), the function / is an isomorphism between X and fX, where the latter carries the relative convexity. By Theorem 3.3, CP functions preserve the median operator (and conversely), showing that fX is median-stable in L. Now let L be a distributive lattice, and let X a L be median-stable. For x, y, z, in X, we find that the sets

hx{x,y}nhx{y,z}nhx{z,x}, h{x, y] n h{y, z) n h{z, x} n X are equal (h denotes the hull operator of L, hx is the relative hull in X). The second set consists only of the median of x,y,z, and this point is in X. Hence the relative convexity of X is binary. The lattice L is S2 by Theorem 3.5(1), and the property S2 is hereditary. The median operator of a distributive lattice has already been considered by BirkhofT and Kiss in [2]. They define the segment joining a to b as

{x: m(a,b,x) = x}. This is exactly the interval h{a,b}. They also observed that m(a,b,c) is the unique point which is in all three intervals h{a,b), h{b,c], h{c,a}, and that for each a,a' in L with h{a,a'} = L there is an induced distributive lattice structure on L, defined by x A y = m(a,x,y),

x v y = m(a',x,y).

This is, in fact, an algebraic formulation for Theorem 3.5(2). 4. Completeness and the intrinsic topology 4.1. Complete sets. Let {X, ^ ) be a partially ordered set. As in [5], a downdirected or updirected set is assumed to be non-empty. A subset C of X is downcomplete in (X, ^ ) provided that for each downdirected set A e C which has a lower bound in X, the infimum of A exists and is in C. In contrast with the usual definitions of 'completeness' (in lattices or in semilattices), the present concept is conditional (the existence of lower bounds) and relative (to the superset). Upcompleteness in {X, ^ ) is


17

defined dually. A set which is both downcomplete and upcomplete in (X, ^ ) will be called complete in (X, ^ ) for short. Now let X be an S3 convex structure. A subset of X is called (up/down) complete in X if it is (up/down) complete in (X, ^b) for each b e X. Note that in (X, ^b), the point b is the smallest element. 4.2. THEOREM. Let X be a topological convex structure with compact intervals, such that each base-point order has a closed graph. Then every closed subset of X is complete inX. Proof. Fix a base-point b, and let A c X be 6-downdirected with infimum u. By 3.1 (1), A is also u-downdirected and u = u-inf/4. By Lemma 3.2(1),

{u}=()h{a,u}. as A

If U is an open neighbourhood of u, then for some a e A, h{a, u} 1 vertices. Let C(i), with i e /, be a descending chain of complete convex subsets of P. The index set / is totally ordered by i ^j

if and only if

C{i) 3 CO').

For each i we let x(i) be the nearest-point of an in C(i). Note that (3)

fori^y,

x{i)eh{an,x(j)}

(or x(i) (2). This follows from the fact that singletons are convex closed. Proof of (2) => (3). Since X is Hausdorff, we find from Theorem 4.9(3) that the intrinsic (that is, interval) topology on each interval is Hausdorff. By [27, Theorem 7], all intervals must be completely distributive as a lattice. Proof of (3) => (1). All intervals are completely distributive, and hence also infinitely distributive. By Lemma 4.11, all nearest-point functions are continuous. By [27, p. 229], all intervals have enough continuous lattice homomorphisms (CP maps) into [0,1] to separate the points. Composing these maps with nearest-point maps onto intervals, we conclude that the points of X can be separated with CP maps X -> [0,1]. Then X is normal by the screening lemma, 2.4. Proof of (4) => (2). A space with a closed partial order is Hausdorff. Proof of (1) => (4). See [30, 2.7]. 4.13. THEOREM. Let X be a complete binary S 4 convex structure with closed basepoint orders, and let C be convex and complete in X. Then the following are equivalent: (1) C is compact in the intrinsic topology of X; (2) for each b e X, if A cz C is b-updirected, then A has an upper bound in C; (3) for some b e C, if A cz C is a b-chain, then A has an upper bound in X. Proof of (1) => (2). Let C be compact, let b e X, and let A c C be 6-updirected. The sets ^b(a)nC,

aeA,

24

M. VAN DE VEL

are closed in C, and have the finite intersection property. Any point in their intersection is an upper bound for A. Statement (3) is just a particular case of (2). Proof of (3) => (1). Assume that C is not compact. Then there is a transfinite decreasing collection of closed convex subsets C, c C with empty intersection. Let a, be nearest in C, to the given point b of C. One easily verifies that the a, form a totally ordered set (chain) under ^ 6 , which has an upper bound d in X. As C is complete in X, the chain has a supremum c in C. As the a, are cofinally in each C,, we obtain from the completeness of C, in X that c is a common point of all C,, a contradiction. As a consequence, a convex set is compact in the T 2 intrinsic topology of X if and only if it is unconditionally complete in X (see also a remark in 4.1). 5. Applications and problems 5.1. THEOREM. Let X be a binary normal convex structure with compact intervals. Then the weak topology of X—which is Hausdorff—equals the intrinsic topology. In particular, all polytopes in X are weakly compact. Proof. By [30, 2.7], each base-point order is closed with respect to the given topology of X. Then by Theorem 4.2, each convex closed subset of X is intrinsically closed in X. Hence, normality of the X-convexity in the original topology implies that every two distinct points of X can be screened with intrinsically closed convex sets. By the screening lemma, 2.4, we find that the convexity of X is normal in the intrinsic topology as well. From [33, 2.4], we now obtain that the weak topologies of the original X and of X-with-intrinsic-topology must be equal. But the intrinsic topology equals its weak topology by construction. The second part of the theorem follows from Theorem 4.9(1). This result confirms our original motivation for studying the intrinsic topology. One other consequence of Theorem 5.1 is the following. Let X be a compact space with a normal binary convexity. Then the given topology of X is exactly the intrinsic topology. It has already been observed in [33, 2.4] that there is at most one compact topology on X turning a binary S 4 convexity on X into a normal one. A necessary and sufficient condition for the existence of such of a unique topology can be obtained from Theorems 4.12 and 4.13: X is unconditionally complete, and each interval is completely distributive in the base-point order of an endpoint. That weak compactness of polytopes can be derived from the compactness of intervals is a new and somewhat surprising result. We note that this fact can also be deduced from the existence of'compactifications' (Theorem 5.4(3) below), in which the original space is embedded as a (dense) convex subspace: a polytope of the subspace will also be a polytope in the compact superspace. 5.2. COROLLARY. Let X, Y be normal binary convex structures with compact intervals, and carrying the weak topology. Let f: X -> Y be injective, CP, and continuous. Then f is an embedding of topological convex structures.


25

Proof. Now f(X) is binary and S 4 in its relative convexity, by Theorem 2.9(4). As / : X -> f(X) is bijective and preserves convexity in both directions, Theorem 2.9(1), we conclude that / is a homeomorphism for the intrinsic topology of X and of f{X). Let 3~r be the relative topology of/(X) and let ^ be the intrinsic topology of f(X). As / is continuous in the given topologies, we find that 3~r Y be a surjective CP function. Then f is weakly continuous if and only if its fibres are closed. Proof. Use Lemma 4.10 and Theorem 5.1. 5.4. THEOREM. Let X be a complete binary S 4 convex structure with a T 2 intrinsic topology. (1) IfC c X is compact convex, and ifO 3 C is open, then there is a convex open set P with C cz P a 0; in particular, X is locally convex. (2) The median operator m: X3 -*• X is intrinsically continuous. (3) X embeds^ as a dense convex subspace of a normal binary convex structure on a compact space. Proof of (1). As the intrinsic topology is generated by the convex closed sets, one can associate to each x e C a finite convex closed covering of X\O leaving x uncovered. As C is compact, this results in a finite number of points x,- 6 C, and for each i, a finite convex closed covering of X\O with sets D^, such that

The left-hand expression can be written in the form \JkEk, where each Ek is the intersection of certain D(J. Hence Ek is convex closed and disjoint with C. As the convexity of X is normal (Theorem 4.12), there exists an open half-space Pk with

CczPk,

PknEk = 0.

The desired convex open set is f]kPk. Proof of (2). For each x,y e X the nearest-point function

t Relative to both its topology and its convexity.

26

M. VAN DE VEL

is continuous (Lemma 4.11) and CP (Theorem 2.8). This leads us to a continuous CP function / : X -> ]~]x..yM;c>);}> which is easily seen to be injective. Each h{x,y} is a completely distributive lattice, and the product convexity (see 1.7) corresponds to the product lattice structure which is again completely distributive [5,1.2.7, VII.2.9]. By Corollary 5.2, / is an embedding. Hence f(X) is a median-stable subset of the product, and its median operator is derived from the one of the product. In a completely distributive lattice, the operations ' A ' and ' v ' are continuous, and hence its median operator is continuous. It follows that m: X3 -> X is continuous. Proof of (3). With the above notation, one can verify that the closure of/(X) is a compact extension of X as required. Let us describe a different and internal method of compactification. The collection of all convex closed subsets of X is a normal Tx subbase (see 1.5) for the intrinsic topology. Let X* be the corresponding superextension. We have an embedding X -> X* as described in 1.5. Note that a maximal linked system of convex closed sets in X is actually an ultrafilter since X is binary. Hence X is dense in X*. If P a X is a polytope and if i f is a maximal linked system with P e H. Suppose H contains a


27

point y0 ^ x. By the continuity of q, H = q(P~) (2). Let x,y e Lbe arbitrary. Then h{x, y} = [x

A

y, x v y]

is a completely distributive sublattice of L, and hence each of its (convexity) intervals is completely distributive in the base-point order of an endpoint. In particular, h{x,y} is completely distributive under '^ x '. Proof of (2) => (3). By Theorem 5.4(3), L embeds as a dense convex subset of a compact normal binary L*. Fix a e L, and put L(a) = {x: x ^ a}, M(a) = {x: x ^ a}. Then L(a) cz L is updirected under ^ a , and as L* is compact, L(a) has an a-supremum, say 0. Similarly, M(a) has an a-supremum 1 in L*. Note that the point a is in h{0,1}: if it is not, then there is a closed half-space H a L* with aeH,

Hn/i{O,l} = 0 .

Then H n Lisa prime (dual) ideal of L, and hence it includes an end of L(a) or of M(a). But as H cz L* is closed, we obtain that 0 G H or that 1 G H, a contradiction. Let x eL be arbitrary. Then x

A

a

G

L(a) cz h{0, a},

x v a

G

M(a) cz h{a, 1},

28

M. VAN DE VEL

and as a e h{0,1}, we find that x e h{x A a,x v a] c Ai{0,1}. As L c= h{0,1} and as L is dense in L*, we conclude that L* = h{0,1}, and hence that L* is a completely distributive lattice under ^ 0 . One easily verifies that ^ 0 agrees with the original order ^ on L. Proof of (3) => (1). This is easy. Finally we pay some attention to functions in several variables, which are continuous in each variable separately. 5.7. THEOREM. Let X, Y be normal binary convex structures with compact intervals, and equipped with the weak topology. Let Z be a compact Hausdorjf space, and let / : Z x X -> Y be continuous in each variable separately, and CP in the second variable. If for each z e Z, the hull of f({z} x X) is dense in Y, then f is continuous. The proof relies on the following general fact. See [4] for terminology on function spaces. LEMMA. Let X be locally polytopal, that is, each point of X has a neighbourhood base consisting of polytopes (in an arbitrary, fixed convexity on X), and let Y be locally convex. If J is a collection of continuous CP functions X -> Y, then the topology of pointwise convergence and the compact-open topology coincide on J.

Proof. Since the compact-open topology is finer than the topology of pointwise convergence [4, p. 207], we have to show only that for a compact K