aa r X i v : . [ m a t h . GN ] D ec A MEMO ON CHAINS AND THEIR TOPOLOGIES(WORK IN PROGRESS)
PAUL PONCETA
BSTRACT . We summarize some facts on chains (totally ordered sets),from an order-theoretic and from a topological point of view. We high-light the fact that many classical theorems that are true for partially or-dered sets under some completeness assumption remain true for chainswithout any kind of completeness.
1. E
VERY CHAIN IS A CONTINUOUS POSET A partially ordered set or poset p P, ďq is a set P equipped with a re-flexive, transitive, and antisymmetric binary relation ď . If P has a least(resp. greatest) element, we customarily denote it by (resp. by ). Aposet is complete (resp. conditionally-complete ) if every subset (resp. everynonempty subset bounded above) has a supremum, in which case every sub-set (resp. every nonempty lower-bounded subset) has an infimum. A posetis up-complete if every nonempty subset has a supremum.A nonempty subset C of P is a chain (or a totally ordered subset ) if, forall x, y P C , one of the relations x ď y and y ď x holds. We write x ă y when x ď y and x ‰ y . A chain M is maximal if, for each chain C in P , C Ą M implies C “ M .A nonempty subset D of a poset P is directed if, for all x, y P D , onecan find z P D such that x ď z and y ď z . We say that x P P is way-below y P P , written x ! y , if, for every directed subset D with a supremum Ž D , y ď Ž D implies x ď d for some d P D . We then say that the poset P is continuous if ÓÓ x : “ t y P P : y ! x u is directed and x “ Ž ÓÓ x , for all x P P . The next result describes the way-below relation on a chain. Lemma 1.1.
Let p P, ďq be a chain. The following implications hold: x ă y ñ x ! y ñ x ď y, for all x, y P P . Date : July 24, 2018.2010
Mathematics Subject Classification.
Key words and phrases. chain, totally ordered set, partially ordered set, intrinsic topol-ogy, order topology, interval topology, Lawson topology, Scott topology, continuous poset,compact pospace. roof. Assume that x ă y and that y ď Ž B , for some B Ă P withsupremum. If b ă x for all b P B , then Ž B ď x ă y ď Ž B , which isnot. Thus, x ď b for some b P B , and x ! y . (cid:3) We write ó x for P z Ò x “ t y P P : y ă x u , and ò x dually. Recall that anelement x of a poset P is compact if x ! x .In [5, Example I-1.7], we read that every complete chain is a continuouslattice, but the following more specific result has not been asserted beforeZhao and Zhou [10]. Theorem 1.2.
Let p P, ďq be a chain. Every point x of P is compact orsatisfies x “ Ž ó x . In particular, P is a completely distributive (or super-continuous) poset, hence also a continuous poset.Proof. Suppose that x is not compact. This means that there exists some B Ăó x with supremum such that x “ Ž B . Let u be an upper bound of ó x . Since B Ăó x , u is also an upper bound of B , so that x “ Ž B ď u .This proves that x is the supremum of ó x . (cid:3) A transitive binary relation ă on a set is idempotent if, for all x, y suchthat x ă y , there exists some z with x ă z ă y . A poset is order-dense ifthe strict order relation ă is idempotent. Remembering that the way-belowrelation of a continuous poset is idempotent, the following result is a clearconsequence of the previous theorem. Corollary 1.3.
Let p P, ďq be a chain. The following conditions are equiv-alent: (1) the relations ă and ! agree (except perhaps at ), (2) P has no compact element (except perhaps ),If these conditions are satisfied, then P is order-dense. Conversely, if P isconditionally-complete and order-dense, then these conditions are satisfied.Proof. Condition (1) clearly implies (2), and the fact that (2) implies (1) isa consequence of Lemma 1.1. By continuity of P , ! is idempotent, hence P is order-dense if (1) is satisfied. Assume now that P is conditionally-complete and order-dense. If x is a compact element and ó x ‰ H , then ó x has a supremum u . Since x is compact, we cannot have u “ x , so that u ă x . But P is order-dense, hence there exists some v with u ă v ă x ,which contradicts the definition of u . Thus, we get ó x “ H , i.e. x is theleast element of P . (cid:3)
2. T
OPOLOGIES AND CONVERGENCE ON CHAINS
Topologies on chains.
Let p P, ďq be a chain. The upper topology ,denoted by ν p P q , is the topology generated by the complements ò x of rincipal ideals (as subbasic open sets). Dually, the lower topology ω p P q is the topology generated by the complements ó x of principal filters. The intrinsic topology i p P q is the join of the lower and the upper topologies. Proposition 2.1.
Let p P, ďq be a chain equipped with its intrinsic topology.Then P is a pospace (the order ď is closed in P ˆ P ), hence Hausdorff,and is a topological lattice.Proof. The aim is to show that the map f : p x, y q ÞÑ x ^ y is continuous.Let G be an open subset of P such that x ^ y P G for some x, y P G with x ă y . Then p x, y q P G ˆ ò x and G ^ ò x Ă G , so we deduce that f ´ p G q is open in P . Dually, the map p x, y q ÞÑ x _ y is continuous, hence P is atopological lattice. (cid:3) In a poset P , we write B Ò (resp. B Ó ) for the subset made up of upperbounds (resp. lower bounds) of the subset B . It is remarkable that the fol-lowing proposition holds even for non-complete chains. Proposition 2.2.
Let p P, ďq be a chain. Then, (1) the upper topology and the Scott topology agree on P , (2) the lower topology and the dual Scott topology agree on P , (3) the intrinsic topology, the interval topology, the open-interval topol-ogy, the order topology, the bi-Scott topology, the Lawson topology,and the dual Lawson topology agree on P .Proof. It is well known that every principal ideal is Scott-closed, hence theScott topology refines the upper topology. Now let F be a Scott-closedsubset of P . We want to show that F “ Ş x P F Ò Ó x , the latter being closedin the upper topology. The inclusion Ă is clear. So let y P Ş x P F Ò Ó x , whichmeans that y is lower than every upper bound of F . Assume that y R F . If f P F , we cannot have y ď f since F “Ó F . Consequently, y ą f , andwe deduce that y is an upper bound of F , and even the least upper bound: y “ Ž F . Since F is Scott-closed and admits F as a directed subset withsupremum, we have Ž F “ y P F , a contradiction. So y P F , and F isclosed in the upper topology.Using the equality between the upper and the Scott topology and [2, The-orem 1], the other assertions of the theorem follow. (cid:3) Remark 2.3.
The proof also shows that the Scott closure operator of a chaincoincides with its Dedekind–Mac Neille closure operator.Following e.g. Mao and Xu [7, Definition 3.2], a poset is called hyper-continuous if, for all x , the set t y : y ă x u is directed and admits x assupremum, where ă is the binary relation defined by y ă x ô x P pÒ y q o ,the interior of Ò y being taken in the upper topology. ao and Xu [7, Proposition 3.5] showed that, for a continuous poset,hypercontinuity is equivalent to the coincidence of the upper and the Scotttopology (see also [5, Theorem VII-3.4] for the case of continuous lattices),which happens to be satisfied for chains as asserted by the previous theorem.Thus, we have the following statement. Corollary 2.4.
Every chain is a hypercontinuous poset.
We now come to a very important result on the intrinsic topology ofchains.
Theorem 2.5.
A chain equipped with its intrinsic topology is a completelynormal (= hereditarily normal) space.Proof.
See e.g. Aló and Frink [2] and Alò [1, Theorem 1]. (cid:3)
Remark 2.6.
Steen [8] proved the stronger result that every chain is a hered-itarily collectionwise normal space.As a corollary, we have that every chain is Tychonoff, i.e. a completelyregular Hausdorff space (recall every normal T space is Tychonoff). Thisfact can also be seen by [5, Proposition IV-3.21], which asserts that ev-ery chain allows an embedding into a cube (that is, a lattice r , s X ) thatpreserves all existing sups and infs. Erné [3] proved a stronger result. De-fine a map f between posets P and Q to be cut-stable if, for all A Ă P , f p A Ò q Ó “ f p A q ÒÓ and f p A Ó q Ò “ f p A q ÓÒ . Then, by [3, Corollary 4.4], achain admits a cut-stable embedding in a power-set, hence is Tychonoff.Note that every cut-continuous map is continuous with respect to intervaltopologies. The following theorem is a bit more precise. Theorem 2.7.
A chain P equipped with its intrinsic topology is a strictlycompletely regular pospace, i.e. (1) P is strictly locally order-convex, i.e. it has a basis of open order-convex neighbourhoods, (2) for all closed lower (resp. upper) subset A and x R A , there existssome continuous order-preserving map f : P Ñ r , s such that f p A q “ t u and f p x q “ (resp. f p A q “ t u and f p x q “ ).Proof. We use Xu’s theorem [9, Theorem 3.3], which applies if every lowersubset of P that is closed in the intrinsic (Lawson) topology is also closed inthe lower topology. But this fact comes from [5, Proposition III-1.6], whichsays that if G “Ó G , then G is dually Lawson open (i.e., open in the intrinsictopology) iff G is dually Scott open (i.e., open in the lower topology). Thisresult is also a consequence of Künzi [6, Corollary 4], since every chain isa topological lattice. (cid:3) emark 2.8. For an example of a completey regular pospace that is notstrictly completely regular, see [6].That every chain be locally order-convex (Item (1) of Theorem 2.7) hasbeen known since Alò and Frink, see [2, Theorem 3]. This latter result and[4, Theorem 1, page 36] are gathered in the next theorem.
Theorem 2.9.
Every open subset of a chain (resp. a complete chain) isthe union, in a unique way, of maximal disjoint open order-convex subsets(resp. disjoint open intervals). R EFERENCES [1] Richard A. Alò. A proof of the complete normality of chains.
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COLE P OLYTECHNIQUE , R
OUTE DE S ACLAY , 91128 P
ALAISEAU C EDEX ,F RANCE , AND
INRIA, S
ACLAY –Î LE - DE -F RANCE
E-mail address : [email protected]@cmap.polytechnique.fr