aa r X i v : . [ m a t h . L O ] M a y ORDER CONVERGENCE AND COMPACTNESS
DOMINIC VAN DER ZYPEN
R´esum´e.
Soit ( P, ≤ ) un ensemble partiellement ordonn´e et soit τ une topologie compactesur P qui est plus fine que la topologie d’intervalles. Alors τ est contenu dans la topologiede convergence d’ordre. Topologies on a given poset
On any given partially ordered set ( P, ≤ ) there are topologies arising from the given order ina natural way (see also [2]). Perhaps the best known such topology is the interval topology .Set S − = { P \ ( x ] : x ∈ P } , and S + = { P \ [ x ) : x ∈ P } where ( x ] = { y ∈ P : y ≤ x } and[ x ) = { y ∈ P : y ≥ x } . Then S = S − ∪ S + is a subbase for the interval topology τ i ( P ) on P .There is another natural way to endow an arbitrary poset ( P, ≤ ) with a topology. We wantto describe this topology in the following.A (set) filter F on ( P, ≤ ) is a nonempty subset of the powerset of P such that − ∅ / ∈ F− U, V ∈ F implies U ∩ V ∈ F− U ∈ F and V ⊇ U imply V ∈ F .(Note that the above concept can of course be defined for arbitrary sets.) For any subset A ⊆ P let the set of lower bounds of A be denoted by A l = { x ∈ P : x ≤ a for all a ∈ A } andthe set of upper bounds by A u = { x ∈ P : x ≥ a for all a ∈ A } . If S is a collection of subsetsof P then we set S l = S { S l : S ∈ S} , similarly set S u = S { S u : S ∈ S} .Let A ⊆ P be a subset of a poset P and y ∈ P . We say that y is the infimum of A if y isthe greatest element of A l and write V A = y . Dually we define the supremum of A , written W A . Note that in general, suprema and infima need not exist.Let F be a filter on a poset P and let x ∈ P . We say that F order-converges to x , insymbols F ˙ → x , if V F u = x = W F l . Note that the principal ultrafilter consisting of thesubsets of P that contain x order-converges to x .Now we are able to define the order convergence topology τ o ( P ) (called order topologyin [1]) on any given poset P by: τ o ( P ) = { U ⊆ P : for any x ∈ U and any filter F with F ˙ → x we have U ∈ F } . AMS subject classification (2000): 06B15, 06B30Keywords: interval topology, order convergence, order topology, compact ordered spaces
It is straightforward to verify that this is a topology. Indeed, τ o ( P ) is the finest topology on P such that order convergence implies topological convergence (which is not hard to proveeither). We will make constant use of the following facts: FACT 1.1.
Let P be a poset, let F be a filter on P . Then: (1) x ∈ F u ⇔ ( x ] ∈ F and x ∈ F l ⇔ [ x ) ∈ F . (2) If F ˙ → x then F u = ∅ 6 = F l . (3) Suppose F ˙ → x . If x a then P \ ( a ] ∈ F . Dually if x b then P \ [ b ) ∈ F . (4) If F ˙ → x and G is a filter on P with G ⊇ F then G ˙ → x .Proof. The proofs of assertions 1 and 2 are straightforward, and assertion 3 follows directlyfrom [1], p. 3. We prove assertion 4. Since G u ⊇ F u it suffices to show that G u ⊆ [ x ) inorder to get V G u = x . Assume that there is y ∈ G u \ [ x ). By assertion 1, ( y ] ∈ G . Since wehave x y , we get P \ ( y ] ∈ F ⊆ G (by assertion 3). So ( y ] ∩ ( P \ ( y ]) = ∅ ∈ G , which is acontradiction. The statement W G l = x is proved similarly. (cid:3) The result
Note that 1.1, assertion 3 implies that for any poset P , the interval topology τ i ( P ) is containedin the order convergence topology τ o ( P ). The following theorem connects the concepts ofinterval topology, order convergence and compactness. THEOREM 2.1.
Let ( P, ≤ ) be a poset. If τ is a compact topology on P such that τ i ( P ) ⊆ τ ,then τ ⊆ τ o ( P ) .Proof. Suppose that W ∈ τ \ τ o ( P ). Then there is x ∈ W and a filter F on P such that F ˙ → x and W / ∈ F .The strategy now is to find an ultrafilter on the closed set Q := P \ W of ( P, τ ) that doesnot converge to any point of Q with respect to the subspace topology of ( P, τ ) on Q . Thiswill imply that Q is a non-compact closed subset of ( P, τ ), which in turn implies that (
P, τ )is not compact.Note that every element of F intersects Q (otherwise we would have W ∈ F ). So F ∪ { Q } is a filter base which is contained in some ultrafilter U . Moreover, by 1.1, assertion 4, theultrafilter U order-converges to x .It is easy to check that U | Q = { U ∩ Q : U ∈ U } is an ultrafilter on Q (this uses of course the fact that Q is a member of U ). Claim:
U | Q does not converge to any y ∈ Q with respect to τ | Q , the topology on Q inducedby τ . Proof of Claim : Pick any y ∈ Q . First, we know that x ∈ W and y ∈ Q , whence x = y .Suppose that the following holds in P :(A) For all z ∈ U u we have y ≤ z and for all z ′ ∈ U l we have y ≥ z ′ . RDER CONVERGENCE AND COMPACTNESS 3
Then by definition of order convergence this would imply y ≤ x , since x = V U u , and similarlywe would get y ≥ x , a contradiction to x = y . So, (A) must be false, and without loss ofgenerality we may assume that there is a z ∈ U u with y z . By 1.1, assertion 1, we get( z ] ∈ U which implies B := ( z ] ∩ Q ∈ U | Q . Since y z we also have y ∈ P \ ( z ] . ( ⋆ )Because τ contains the interval topology τ i ( P ), statement ( ⋆ ) above implies that the set V := ( P \ ( z ]) ∩ Q = Q \ B is an open neighborhood of y in ( Q, τ | Q ). But since B ∈ U | Q and V = Q \ B , we have V / ∈ U | Q , so U | Q does not converge to y with respect to τ | Q . Since y ∈ Q was arbitrary, theclaim is proved.The claim now shows that Q = P \ W is a closed, non-compact subset of ( P, τ ). So (
P, τ )cannot be compact. (cid:3)
This theorem has a direct consequence for Priestley spaces, i.e. compact totally order-disconnected ordered spaces as introduced in ([3], [4]).
COROLLARY 2.2. If ( P, τ, ≤ ) is a Priestley space, then τ ⊆ τ o ( P ) . Acknowledgement:
The author wishes to thank Jana Flaˇskov´a for pointing out a mistakein the proof of fact 1.1 and providing a more direct argument.
References [1] M.Ern´e,
Topologies on products of partially ordered sets. III. Order convergence and order topology , Alge-bra Universalis (1981), no. 1, 1–23.[2] G. Gierz, H. Hofmann, K. Keimel, J. Lawson, M. Mislove, D. Scott: A Compendium of ContinuousLattices , Springer-Verlag, 1980.[3] H.A.Priestley,
Representation of distributive lattices by means of ordered Stone spaces , Bull. London Math.Soc. 2 (1970), 186–190.[4] H.A.Priestley,
Ordered topological spaces and the representation of distributive lattices , Proc. LondonMath. Soc. (3) (1972), 507–530.D. van der ZypenAllianz Suisse Insurance CompanyBleicherweg 19, CH-8022 Zurich, Switzerland(1972), 507–530.D. van der ZypenAllianz Suisse Insurance CompanyBleicherweg 19, CH-8022 Zurich, Switzerland