Chapter IX Theory of Convergence 1. Introduction In Chapters II and III, we discussed the convergence of sequences. Sequential convergence is very useful in working with first countable spaces (in particular, with metric spaces). In any space \, if Ð+Ñ88is a sequence in E and Ð+ÑÄB, then B−cl EÞ But in a first countable space, the converse is also trueÀ B − cl Eiff there is a sequence Ð+88 Ñ in E with Ð+ Ñ Ä BÞ Therefore convergent sequences determine the set clE , so sequences can be used to check whether of not cl EœE, that is, whether or not E is closed. we can determine whether of not Eis closed using sequences. (See Theorem III.9.6 ). So in a first countable space “sequences determine the topology.” However, sequences are not sufficient in general to describe closures: for example, in Ò!ß=" Ó we have ==""−cl Ò!ß Ñ but no sequence Ð α= 8" Ñ in Ò!ß ß Ñconverges to = ". The basic neighborhoods Ð α= ß " Ó of =" are very nicely ordered, but there are just “too many” of them: no mere sequence ÐÑα=8" in Ò!ßÑ is “long enough” to be eventually inside every neighborhood of =". A second example (see Example III.9.8 ): Ð!ß !Ñ −cl ÐP ÖÐ!ß !Ñ×Ñ but no sequence in ÐP ÖÐ!ß !Ñ×Ñ converges to Ð!ß !Ñ. Here the problem is different. Pis a “small” space ( lPl œ i! ) but the neighborhood system aÐ!ß!Ñ ÐÑordered either by inclusion or by reverse inclusion is a very complicated poset so complicated that a mere sequence Ð+8 Ñ in P ÖÐ!ß !Ñ× cannot eventually be in every neighborhood of Ð!ß !ÑÞ In this chapter we develop a theory of convergence that is sufficient to describe the topology in any space \Þ0−\. We define a kind of “generalized sequence” called a net A sequence is a function , A and we write 0Ð8ÑœB8 Þ A net is a function 0−\, where ÐA ß ŸÑis a more general kind of ordered set. Informally, we write a sequence 0ÐBÑ as 8 ; similarly, when 0 is a net, then we informally write the net as ÐB- ÑÞ Thinking of \œÒ!ß=" Ó, we might hope that it would be a sufficient generalization to replace with an initial segment of ordinals AαœÒ!ß Ñ: in other words, to replace sequences with “transfinite sequences” with domain some well-ordered set “longer” than . In fact, this is a sufficient generalization to deal with a case like Ò!ß=="" Ó À if ÐBα Ñ is the “transfinite sequence” in Ò!ß Ñ given by Bœαααα Ð ="" Ñ, then ÐBÑÄ = Ðin the sense that ÐBÑ αis eventually in every neighborhood of =" Ñ. But as we will see below, such a generalization does not go far enough (see Example 2.9 ). We need a generalization that uses some kind of ordered set Aα more complicated than just initial segments Ò!ß Ñ of ordinals. The theory of nets turns out to have a “dual” formulation in the theory of filters. In this chapter we will discuss both formulations. It turns out that nets and filters are fully equivalent formulations of convergence, but sometimes one is more natural to use than the other. 2. Nets 388 Definition 2.1 A nonempty ordered set ÐߟÑA is called a directed set if 1) Ÿ is transitive and reflexive 2) for all --"#, −− A, there exists - $ A such that - $ -" and - $ - #. Example 2.2 1) Any chain (for example, ) is a directed set. 2) In ‘‘, define BŸ‡‡‡ C if lBl lClÞ Then Ð, Ÿ Ñ is a directed set and BŸ C means that “ B is at least as far from the origin as C"Ÿ""Ÿ"Ÿ.” Notice that ‡‡ and , so that ‡ is not antisymmetric. A directed set might not be a poset. 3) Let aB be the neighborhood system at B\ß in ordered by “reverse inclusion” that is, RŸR"# iff RªRÞ "#B a Ág and condition 1) in the definition clearly holds. Condition 2) is satisfied because for RßR"# −aa B ßwe have R " ∩R # œR $ − Band R $ R " and R $ R2 . Therefore ÐߟÑaB is a directed set. This example hints at why replacing A (in the definition of a sequence) with a directed set (in the definition of net) will give a tool strong enough to describe the topology in any space \À the directed set A can chosen to be as complicated as the most complicated system of neighborhoods at a point B. 4) Let A œÖJ©Ò!ß"ÓÀ!−Jß"−J and J is finite ×Þ In analysis, J is called a partition of the interval Ò!ß "Ó. Order A by inclusion: a “larger” partition is a “finer” one one with more subdivision points. Then A is a directed set. Definition 2.3 i) A net in a set \ is a function 0ÀÐAA ߟÑÄ\, where Ð ßŸÑ is directed. We write 0Ð- Ñ œ B-- and, informally, denote the net by ÐB Ñ. ii) A net ÐB-- Ñ in a space \ converges to B − \ if ÐB Ñ is eventually in every neighborhood of Bthat is, aR−a-AB! b − such that B- −R wherever -- !. iii) A point B\ in a space is a cluster point of a net ÐBÑ- if the net is frequently in every neighborhood of BR−− , that is: for all a-A--B! and all there is a ! for which B−R- . Clearly, every sequence is a net, and when Aœ , the definitions ii), and iii) are the same as the old definitions for convergence and cluster point of a sequence. Example 2.4 A---œÖßß×"#$ and define an order in which -- $" and -- $# but - " and - # are not comparable. ÐߟÑAA‘- is a directed set Þ We can define a net 0À Ä by 0ÐÑœ!$ and assigning any real values to 0Ð--"# Ñ and 0Ð Ñ. Since B-- − Ð %%-- ß Ñ for all $, we have ÐB Ñ Ä !Þ ()Notice that a net can have a finite domain and can have a “last term.” 389 Example 2.5 The following examples indicate how several different kinds of limit that come up in analysis can all be reformulated in terms of net convergence. In other words, many different “limit” definitions in analysis are “unified” by the concept of net convergence. 1) Let A‘œ Ö+× and define BŸ‡ C (in A) iff ±B+± ±C+± (in ‘). Suppose 0ÀA‘ Ä is a net. Then the net ÐBÑÄP−- ‘ ‡ iff for all %-A%%--! there exists !! −such that B- −ÐP ßP Ñ if iff for all %-A%%--! there exists ! −such that B- −ÐP ßP Ñ if !l +lŸl! +l iff for all %$! there exists !such that l0Ð -%-$ ÑPl if !l +l ÅÐÊÀuse $-œl! +lÎ# Ñ iff lim 0ÐBÑ œ P (in the usual sense of analysis). BÄ+ 2) Let A‘œŸ0ÀÄ0 with the usual order . If ‘ ‘, we can think of as a net, and write 0Ð<Ñ œ B<< Þ Then the net ÐB Ñ Ä P − ‘ iff for all %A‘%%! there exists <!< − œsuch that B −ÐP ßP Ñ if < < ! iff for all %‘! there exists <!! −such that | 0Ð<ÑPl % if < < iff lim 0ÐBÑ œ P (in the usual sense of analysis). BÄ∞ Parts 1) and 2) show how two different limits in analysis can each be expressed as convergence of a net. In fact each of the limits lim0ÐBÑ œ P, lim 0ÐBÑ œ P, lim 0ÐBÑ œ P, lim 0ÐBÑ œ Pß BÄ+BÄ+ BÄ+ BÄ∞ and lim 0ÐBÑ œ P can be expressed as the convergence of some net. In each case, the trick is to BÄ∞ choose the proper directed set. 3) Integrals can also be defined in terms of the convergence of nets. Let 1 be a bounded real valued function defined on Ò!ß "Ó and let A œ ÖJ © Ò!ß "Ó À J is finite and !−Jß"−J×. Order A by inclusion: J"# ŸJ iff J "# ©J iff J # is a “finer” partition than JÞ " For J −A?, enumerate J as !œB!" B ÞÞÞB 8"8 B œ"Þ Let B 333" œB B and 8 inf . 0ÐJÑ œ Ð 1 ц?‘ B3J œ B − 3œ" ÒB3" ß B 3 Ó 0ÀA‘ Ä is a net in ‘ and we can write 0ÐJÑœBJ ÞThen ÐBJ! Ñ Ä P iff for all % ! there is a partition J such that for all partitions JJßlBPl finer than !J % One can show that such an P always exists: P is called the lower integral of 1 over Ò!ß "Óß denoted Pœ" 1. ! If we replace “inf” by “sup” in defining 0Y, then the limit of the net is called the upper integral of 1 " over Ò!ß "Ó, denoted Y œ 1. If P œ Y, we say that 1 is (Riemann) integrable on Ò!ß "Ó and write ! "1œPÐœYÑÞ ! 390 A sequence can have more than one limit: for example, if l\l " and \ has the trivial topology, every sequence ÐB8 Ñ in \ converges to every point in \. Since a sequence is a net (with A œ ), a net can have more than one limit. We also proved that a sequence in a Hausdorff space can have at most one limit (Theorem III.9.3). This theorem still holds for nets, but then even more is true: uniqueness of net limits actually characterizes Hausdorff spaces. Theorem 2.6 A space \\ is Hausdorff iff every net in has at most one limit. Proof Suppose \ is Hausdorff and BÁC−\. Choose disjoint open sets YßZ with B−Y and C−ZÞ If a net ÐBÑÄB--, then ÐBÑ is eventually in Y and therefore ÐBÑ - is eventually outside Z.