Chapter VIII Ordered Sets, Ordinals and Transfinite Methods 1. Introduction In this chapter, we will look at certain kinds of ordered sets. If a set \ is ordered in a reasonable way, then there is a natural way to define an “order topology” on \. Most interesting (for our purposes) will be ordered sets that satisfy a very strong ordering condition: that every nonempty subset contains a smallest element. Such sets are called well-ordered. The most familiar example of a well-ordered set is and it is the well-ordering property that lets us do mathematical induction in In this chapter we will see “longer” well ordered sets and these will give us a new proof method called “transfinite induction.” But we begin with something simpler. 2. Partially Ordered Sets Recall that a relation V\ on a set is a subset of \‚\ (see Definition I.5.2 ). If ÐBßCÑ−V, we write BVCÞ An “order” on a set \ is refers to a relation on \ that satisfies some additional conditions. Order relations are usually denoted by symbols such asŸ¡ß£ , , or . Definition 2.1 A relation V\ on is called: transitive ifÀ a +ß ,ß - − \ Ð+V, and ,V-Ñ Ê +V-Þ reflexive ifÀa+−\+V+ antisymmetric ifÀ a +ß , − \ Ð+V, and ,V+ Ñ Ê Ð+ œ ,Ñ symmetric ifÀ a +ß , − \ +V, Í ,V+ (that is, the set V is “symmetric” with respect to thediagonal ? œÖÐBßBÑÀB−\ש\‚\). Example 2.2 1) The relation “œ\ ” on a set is transitive, reflexive, symmetric, and antisymmetric. Viewed as a subset of \‚\, the relation “ œ ” is the diagonal set ? œÖÐBßBÑÀB−\×Þ 2) In ‘, the usual order relation is transitive and antisymmetric, but not reflexive or symmetric. 3) In ‘, the usual order Ÿ is transitive, reflexive and antisymmetric. It is not symmetric. 4) On any set of cardinal numbers V we have a relation Ÿ . It is transitive, reflexive and antisymmetric (by the Cantor-Schroeder-Bernstein Theorem I.10.2 ), but not symmetric Ðllœ"ÑÞunless V 319 Definition 2.3 A relation Ÿ\ on a set is called a partial order if Ÿ is transitive, reflexive and antisymmetric. The pair Ð\ß Ÿ Ñ is called a partially ordered set (or, for short, poset). A relation Ÿ\ on a set is called a linear order if Ÿ is a partial order and, in addition, any two elements in \ are comparable: a+ß,−\ either +Ÿ, or ,Ÿ+. In this case, the pair Ð\ߟ) is called a linearly ordered set. For short, a linearly ordered set is also called a chain. We write +, to mean that +Ÿ, and +Á,ÞFor any sort of order relation Ÿon \, we can invert the order notation and write , + Ð,+Ñto mean the same thing as +Ÿ, Ð+,ÑÞ In some books, a partial order is defined as a “strict” relation which is transitive and irreflexiveÐa+ − \ , a Î +ÑÞ In that case, we can define +Ÿ, to mean “ +, or +œ,” to get a partial order in the sense defined above. This variation in terminology creates no real mathematical problems: the difference is completely analogous to worrying about whether “Ÿ ” or “ ” should be called the“usual order” on ‘ . Example 2.4 1) Suppose E©\ÞFor any kind of order Ÿon \ß we can get an order ŸE on E©\ by restricting the order ŸEÞ to More formally, ŸœŸ∩E‚EE ( ). We always assume that a subset of an ordered set has this natural “inherited” ordering unless something else is explicitly stated. With that understanding, we usually omit the subscript and also write ŸŸE for the order E on . If Ð\ß Ÿ Ñ is a poset (or chain), and E © \, then ÐEß Ÿ Ñ is also a poset (or chain). For example, every subset of Ðߟё is a chain. 2) For Dß A −‚ Ð œthe set of complex numbers Ñ , define D £ A iff lDl Ÿ lAlß where Ÿ is the usual order in ‘‚. Ðᜥ is not a poset. (Why? ) 3) Let Ð\ßg Ñ be a topological space. For 0ß 1 − GÐ\Ñ, define 0Ÿ1 iff aB−\, 0ÐBÑŸ1ÐBÑ. As a set, Ÿ œ ÖÐ0ß 1Ñ − GÐ\Ñ ‚ GÐ\Ñ À aB −‘ ß 0ÐBÑ Ÿ 1ÐBÑ×. Notice that, in contrast to part 2), we are allowing ourselves an ambiguity in the notation here because we are using “ŸŸGÐ\Ñ ” with two different meanings: we are defining an order “ ” in , but the comparison “0ÐBÑ Ÿ 1ÐBÑ ” refers to the usual order a different ordered set, ‘. Of course, we could be more careful and write 0£1for the new order on GÐ\Ñ, but usually we won't be that fussy when the context makes clear which meaning of “Ÿ ” we have in mind. ÐGÐ\Ñß Ÿ Ñ is a poset but usually not a chain: for example, if \ œ‘ and 0ß 1 are given by 0ÐBÑ œ B and 1ÐBÑ œ B#, then 0 ŸÎ1 and 1ŸÎ 0. When is ÐGÐ\Ñß Ÿ Ñ a chain? ÐThe answer is not “ iff l\l Ÿ "Þ” Ñ 320 4) The following two diagrams represent posets, each with 5 elements. Line segments upward from B to C indicate that BŸC. In Figure (i), for example, .Ÿ-and -Ÿ+(so .Ÿ+); in Figure (i), +, and are not comparable. Figure (ii) shows a chain: +Ÿ,Ÿ-Ÿ.Ÿ/Þ 5) Suppose VV is a collection of sets. We can define ŸEŸFE©F on by iff . Then ÐߟÑVV is a poset. In this case, we say that has been ordered by inclusion. In particular, for any set \, we can order cc Ð\Ñby inclusion. What conditions on \ will guarantee Ð Ð\Ñß Ÿ ) is a chain ? 6) Suppose VV is a collection of sets. We can define ŸEŸFEªF on by iff . Then ÐߟÑVV is a poset. In this case, we say that has been ordered by reverse inclusion. In particular, for any set \Ð\Ñ, we can order c by reverse inclusion. What conditions on \ will guarantee that ÐÐ\Ñߟc ) is a chain ? For a given collection V, Examples 5) and 6) are quite similar: one is a “mirror image” of the other. The identity map 3ÀVV Ä is an “order-reversing isomorphism” between the posets. For our purposes, the “reverse inclusion” ordering on a collection of sets will turn out to be more useful. For a point B\ in a topological space , we can order the neighborhood system aB by reverse inclusion ŸÐߟÑ\. The “order structure” of the poset aB reflects some topological properties of and indicates just how complicated the “neighborhood structure” at B is. For example, i) If B is isolated, then ÖB×−RBB and RŸÖB× for every R−R . So the poset ÐߟÑaB has a largest element. ÐIs the converse true? Ñ ii) If \ is first countable and UB"#5 œ ÖR ß R ß ÞÞÞß R ß ÞÞÞ× is a countable neighborhood base at BR−5, then for every aB there is a such that RŸR5B. Thus the poset Ða ßŸÑ contains a countable subset ÖR"# ß R ß ÞÞÞß R 5 ß ÞÞÞ× whose members become “arbitrarily large” in the poset. iii) The poset ÐߟÑaB is usually not a chain. But it does have interesting 321 order property: for any RßR−"#aa B ßbR− $ B such that RŸR " $ and RŸR # $ (let RœR∩R$"#ÑÞ It will turn out (in Chapter 9) that this poset ÐߟÑaB is the inspiration for defining the notion of “convergent nets,” a kind of convergence that is more powerful than “convergent sequences.” Unlike sequences, convergent nets will be able to determine closures (and therefore the topology) in any topological space. Definition 2.5 Suppose Ð\ß Ÿ Ñ is a poset and let E © \Þ An element B − \ is called i) the largest (or last) element in \CŸB if for all C−\ ii) the smallest (or first) element in \BŸC if for all C−\ iii) a maximal element in \ if ÐC B ÊCœBÑ for all C−\ iv) a minimal element in \ if ÐCŸB ÊCœBÑ for all C−\. It is clear that a largest element in Ð\ß Ÿ Ñ, if it exists, is unique. (If DD"# and were both largest, then DŸD"# and DŸD #""# so DœD.) In Figure (i): both +ß , are maximal elements and .ß / are minimal elements. This poset has no largest or smallest element. Suppose a poset Ð\ß Ÿ Ñ has a unique maximal element D. Must D also be the largest element in Ð\ß Ÿ Ñ ? In Figure (ii): +/ is the smallest (and also a minimal element); is the largest (and also a maximal) element. v) Suppose E©\ and B−\ÞWe say that B−\ is an upper bound for E if +ŸB for all +−EàB is called aleast upper bound (sup) for E if B is the smallest upper bound for EÞ The set E\Emight have many upper bounds, one upper bound, or no upper bounds in . If has more than one upper bound, E\\ might or might not have a least upper bound in . But if has a least upper boundB , then the least upper bound is unique (why? ). vi) An element B−\ is called a lowerbound for E if BŸ+ for all +−EàB is called a greatest lower bound (inf) for EB if is the largest lower bound for EÞ The set E might have many lower bounds, one lower bound, or no lower bounds in \E. If has more than one lower bounds, E\\ might or might not have a greatest lower bound in , but if has a greatest lower bound B, then the greatest lower bound is unique (why? ). vii) If BC−\ and if bD−\Î with BDC, then B is called an immediate predecessor of CC and is called an immediate successor of B.