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+Ñ88 is 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 E is 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 Ñ converges 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. P is a “small” space ( lPl œ i! ) but the neighborhood system aÐ!ß!Ñ ÐÑ ordered 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 ŸÑ is 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 = Ð in the sense that ÐBÑ α is 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 ß we have R " ∩R # œR $ − B and 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 B that 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
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 Þ Then
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 . Therefore ÐB- Ñ does not also converge to C .
Conversely, suppose \BÁC−\Y∩ZÁg is not Hausdorff. Then there are points such that ww whenever Y−aaAaaABC and Z− Þ Let œ BC ‚ and order by defining ÐYßZÑŸÐYßZÑ iff Y©Yww and Z©Z (reverse inclusion in both coordinates). For each ÐYßZÑ−A , let 0ÐÐY ß Z ÑÑ œ any point BÐY ßZ Ñ − Y ∩ Z Þ We claim that ÐB ÐY ßZ Ñ Ñ Ä B and ÐB ÐY ßZ Ñ Ñ Ä C .
ww If Z−a-C! , let œÐ\ßZÑ− A . Then if - œÐYßZÑ - ! , we have ww BœB--ÐYww ßZ Ñ −Y∩Z©\∩ZœZ. Therefore ÐBÑÄC . The proof that ÐB- Ñ Ä B is similar. ñ
‡ Example 2.7 The implication ÐÉÑ in Theorem 2.6 is false for sequences. Let \œÒ!ß==" Ó∪Ö" × ‡ where =="  Ò!ß"" ÓÞ Let points in Ò!ß = Ó have their usual neighborhoods and let a basic neighborhood ‡‡ of =α===α="" be a set of the form Ðß"" ÓÖ×∪Ö× , where " Þ (In effect, the basic neighborhoods ‡‡‡ of =="" and """ are identical except for replacing == by or vice versa; it is as if = is a “shadow” of =="" which can't be separated from ).
This space is not Hausdorff, but we claim that a sequence ÐÑ\α8 in has at most one limit.
If Ðαα=αα=88"8" Ñ contains infinitely many terms 55 , then let œ sup Ö À 5 œ "ß #ß ÞÞÞ× , so
that the subsequence ÐÑα88885 is in Ò!ßÓ α . If ÐÑÄ α"α# and ÐÑÄ , then ÐÑÄ α5 " and
Ðα#85 Ñ Ä . Therefore "# , are both in the closed set Ò!ß α Ó . But Ò!ß α Ó is X# so a limit for
ÐÑα"#85 must be unique: œÞ
If only finitely many α=8" 's are less than , we can assume without loss of generality that for all ‡ ‡ 8−Öß×Þ, α==8"" It is easy to see that if α= 8" œ for only finitely many 8 , then ÐÑÄ α 8 =" ‡ only. Similarly, if α=88"8"œ8ÐÑÄÞœ" for only finitely many , then α = only If α= and ‡ α=88œ8ÐÑ" each for infinitely many , then α has no limit.
The next theorem tells us that nets are sufficient to describe the topology in any space \Þ
Theorem 2.8 Suppose E©Ð\ßg Ñ . Then B− cl E iff there is a net Ð+Ñ-- in E for which Ð+ÑÄBÞ
Proof Suppose Ð+ÑÄB-- , where Ð+Ñ is a net in E . For each R−aB , Ð+Ñ - is eventually in R so R∩EÁgÞ Therefore B− cl EÞ
391 We can prove the converse because we can choose a directed set “as complicated as” the neighborhood system of BB−Eœ : Suppose cl and let AaB , ordered by reverse inclusion. For -A-œR− ßlet 0Ð Ñœ+- œ a point chosen in R∩EÞ If -!! œR − A-- and œR ! then +-- −R∩E©R!! ∩E©R . Therefore Ð+ÑÄBÞ ñ
Example 2.9 In general, “sequences aren't sufficient” to describe the topology of a space \ . However we might ask whether something simpler than nets will do. For example, suppose we consider “transfinite sequences”0ÀÒ!ßÑÄ\ that is, very special nets αα , where is some ordinal. It turns out that these are not sufficientÞ
‡ Suppose \ œ Ò!ß=="! Ñ ‚ Ò!ß Ñ © X œ Ò!ß == "! Ó ‚ Ò!ß ÓÞ Let : œ the “upper right corner point” ‡ Ðß=="! ÑÞ Then :−\ß cl but we claim that no transfinite sequence in \ can converge to :−X .
Let 0 À Ò!ßα- Ñ Ä \ and write 0Ð Ñ œ ÐB-- ß C Ñ . We claim that ÐÐB -- ß C ÑÑ ÄÎ: Ðno matter how large an α we use! ÑÞ So ÐÐB-- ß C ÑÑ Ä : À
ranÐ0Ñ must be uncountable.
If ranÐ0Ñ were countable, then the set GœÖBÀ- -α × would be countable and "-α="=œÖBÀ×Þsup- "! Then ran Ð0Ñ©Ò!ßÓ‚Ò!ßÑ and so ÐBßCÑ-- could not converge to : .
For every countable set I©Ò!ßαα Ñ , sup I :
Since II well-ordered, represents an ordinal "α Ÿ , so there is an order isomorphism 1ÀÒ!ß""α ÑÄI for some Ÿ . Certainly sup IŸα Þ If supÐIÑ œα" , then 0 ‰ 1 À Ò!ß Ñ Ä \ would be a transfinite sequence with countable range converging to :0 (since does). But we have shown that a net converging to : must have uncountable range. Therefore sup Iα Þ
For each 7=-α-!7 ß let I œÖ ÀC- œ7× and 7 œ sup I 7 Ÿ α Ð “ Ÿ ” is the most we can say since IßI7 Þ7 might be uncountable). For each "α" is in some 77 so, for that , - "
Therefore supÖÀ7לÞ-=α7! By the preceding paragraph, we conclude -α 7! œ for some 7Þ!
Since ÐBßCÑÄ:, we conclude that ÐBßCÑ Ä:Þ But ÐBßCÑ Ä (= ß7 ), so this -- ---−I77!! --- −I " is impossible. ñ
The definition of a subnet is analogous to the definition of a subsequence (see Definition III.10.1 ).
392 Definition 2.10 Let AA9A and Q0ÀÄ\ÀQÄ be directed sets Suppose , . Suppose that for each -A!!−−QÐÑ , there exists a . such that 9.- !! whenever .. ,
then 0‰99.-9. is called a subnet of 0 . We write Ð Ñœ.- and 0Ð Ð ÑÑœB. Þ Informally, the subnet is ÐB-.. Ñ . The definition of subnet guarantees that 9. Ð Ñ œ - -! whenever . .! Þ
Note: an alternate definition for subnet is used in some books. It requires that
1) 9..9.9. is increasing : if "#Ÿß then ÐÑŸÐÑ " # and
2) 9A-A. is cofinal in : for each !!−−QÐÑ Þ , there is a for which 9.- !!
A subnet in the sense of this definition is also a subnet in the sense of Definition 2.10, but the definitions are not equivalent . Definition 2.10 is “more generous” it allows more subnets because Definition 2.10 does not require 9 to be increasing. For most purposes, the slight disagreement in the definitions doesn't matter. However, the full generality of Definition 2.10 is required to develop the full duality between nets and filters that we discuss later.
We will state the following theorem, for now, without proof. The proof will be easier after we talk about filter convergence. For now, we simply want to use the theorem to highlight an observation in Example 2.12.
Theorem 2.11 In Ð\ßg Ñß if B is a cluster point of the net ÐB-- Ñ , then there is a subnet ÐB. Ñ Ä BÞ
Proof See Corollary 4.8 later in this chapter.
Example 2.12 (See Example III.9.8 ) In PœÖÐ7ß8ÑÀ7ß8 are nonnegative integers × , all points except Ð!ß !Ñ are isolated. Basic neighborhoods of Ð!ß !Ñ are sets containing Ð!ß !Ñ and “most of the points from most of the columns” (where “most” means “all but finitely many”).
393 Let ÐB8 Ñ be an enumeration of P ÖÐ!ß !Ñ× . Although Ð!ß !Ñis a cluster point of ÐB8 Ñ , we proved in Example III.9.8 that no sequence in P ÖÐ!ß !Ñ× can converge to Ð!ß !Ñ . In particular, no subsequence of ÐB8 Ñ can converge to Ð!ß !Ñ . However, Theorem 2.11 implies that there is a subnet of ÐB8 Ñ that converges to Ð!ß !ÑÞ
This might seem surprising, but it simply highlights something is actually clear from the definition of subnet: even if a net is a sequence (AœQ ), the directed set in the definition of subnet need not be . Therefore a subnet of a sequence might not be a sequence.
3. Filters
\\\ is the set of all sequences in , but the “set of all nets in ” makes no sense in ZFC. We can put an order ŸÐߟÑÀ on any set AA to create a directed set for example (using Zermelo's Theorem) we could let ŸÐßŸÑ be a well-ordering of AA . Therefore there at least as many directed sets as there are sets . The “set” of all nets in “ÐA ßŸÑ is a directed set ” is “too big” to be a AA\œ Ö\ ÀÐ ßŸÑ × set in ZFC. This is not only an aesthetic annoyance; it is also a serious set-theoretic disadvantage for certain purposes.
Therefore we look at an equivalent way to describe convergence, one which is strong enough to describe the topology of any space \ but which doesn't have this set-theoretic drawback. The theory of filters does this, and it turns out to be a theory “dual” to the theory of nets that is, there is a natural “back-and-forth” between nets and filters that converts each theorem about nets to a theorem about filters and vice-versa.
Definition 3.1 A filter Y in a set \\ is a nonempty collection of subsets of such that
i) gÂY ii) Y is closed under finite intersections iii) If J−YY and KªJ , then K− .
A nonempty family UY© is called a filter base for Y if Y œÖJ©\ÀJ ªF for some F− U }.
Assuming, as we have stated, that filters will provide an equivalent theory of convergence in a space \\, we see that there is no longer a set-theoretic issue. The set of all filters in makes perfectly good sense: each filter Ycc− Ð Ð\ÑÑ , so the set of all filters in \ is just a subset of cc Ð Ð\ÑÑ .
In a topological space \Àß , there is a completely familiar example of a filter aB the neighborhood system at B . A base for this filter is what we have been calling a neighborhood base, UaBB . In fact, and UB are what inspired the general definitions of filter and filter base in the first place.
If we start with a filter YUY , then a base for a filter must have certain properties: U is nonempty (or else YUœg ), and each F− is nonempty (or else g− Y ). Moreover, if F"# , F − UY © , then F∩F−"#YU so (by definition of a base) F∩FªF "# $ for some F− $ Þ
If, on the other hand, we start with any nonempty collection U of nonempty sets in \ such that
394 whenever FF−"# ,UU , there is a F− $ such that F∩FªF " # $ (*) and define YUYUYYœÖJÀJªF for some F− × , then is a filter and is a base for (check! ). is called is called the filter generated by U ; it is the smallest filter containing U .
If we begin with a nonempty collection f of nonempty sets in \ with the finite intersection property, then
UfœÖFÀF is a finite intersection of sets from × has property (*) so UY is a base for a filter called the filter generated by f ; it is smallest filter containing f .
There can be many neighborhood bases UaBB for a neighborhood system in a space \ . Similarly, a filter YUY can have many different filter bases . In particular, notice that itself is a filter base but usually we want to choose the simplest base possible.
Definition 3.2 A filter base UaU converges to B−\ if for every R−B , there is a F− such that F©RÞ In this case we write UY ÄBÞ (Since a filter is also a filter base, we have also just defined the meaning of Y ÄB .)
If YUUYa is the filter generated by , then clearly ÄB iff ª B . (**)
The following theorem is very simple. It is stated explicitly just to make sure that there are no confusions.
Theorem 3.3 Let Y be a filter in the space \B−\ . For any , the following are equivalent:
i) Y ÄB ii) Yaª B iii)YUU has a base where ÄB . iv) Every base UYww for satisfies UÄBÞ
Proof From Definition 3.2 and the observation (**) shows that i)ÍÍ ii) iii).
iii)ÊÄBR−ß iv) Suppose UYU is a base for and that . Then for each aB there is a F−UUYY for which RªF . If w is another base for then, since F− , F contains some set Fww −UU Þ Therefore RªFªF w ß so w ÄBÞ ñ
iv)Êñ i) because YY is a base for .
Example 3.4
1) Let B−Ð\ßgU Ñ . If BBB is a neighborhood base at B , then U ÄBÞ In particular, a ÄBÞ
2) Suppose YY is a filter in \E©\E∩JÁgJ− . If and for all , then the collection Yf∪ÖEל is a nonempty collection with the finite intersection property. Therefore f generates a
395 filter YYw ªEÂE∩JÁgJ− . So: if Y and for all Y , we can enlarge Y to a filter Yw that also contains the set E .
3) If B−\!! , then YYaY œÖE©\ÀB−E× is a filter. Since ª B!! , we have ÄB . The simplest base for YU is œ ÖÖB!! ×× and U Ä B .
w Suppose FÂYY . Then B! ÂF so cannot be enlarged to a filter Y that also contains the w set F such an Y would contain both ÖB!! × and F , and therefore also contain g œ ÖB × ∩ F , which is impossible. (This also follows from Theorem 3.5, below. )
Therefore, Y is a maximal filter in \ . A maximal filter is called an ultrafilter .
For each B−\ß!!Y œÖE©\ÀB−E× is called a trivial ultrafilter . In general, it takes some more work (and AC) to decide whether a set \ also contains “nontrivial” ultrafilters that is, ultrafilters not of this form. (See Theorem 5.4 and the examples that follow ).
The next theorem gives us a simple characterization of ultrafilters. It is completely set-theoretic, not topological.
Theorem 3.5 A filter YYY in \E©\ßE−\E−Þ is an ultrafilter iff for every either or
Proof ÐÊÑIf E©\ and EÂYY , then EªJ is false for every J − , so J∩Ð\EÑÁg for all J−YfY . Therefore the collection œ ∪Ö\E× has the finite intersection property, so f generates a filter YYwwªœ\E−Þ . So if Y is an ultrafilter, we must have YY and therefore Y ÐÉÑ Suppose that for every E©\ß either E or \E is in YY . If w is a filter and YYwwªœÀE−\E−, then YY for if YY w , then Y which would mean E∩Ð\EÑœg−Y w Þ ñ
Note: The proof ÐÊÑ shows that if a filter YY contains neither E nor \E , then can be enlarged to a new filter Y w containing either E\E or whichever of the two you wish.
Definition 3.6 A point B−\ is a cluster point for a filter base Ua if R∩FÁg for every R− B and every F−UY . (Since a filter is also a filter base, we have also just defined a cluster point for a filter Y.)
It is immediate from the definition that BB is a cluster point of a filter base U iff is in the closure of each set in , that is, if and only if cl . UUB− Ö FÀF− × Clearly, if UUÄB , then B is a cluster point of (explain! ).
Theorem 3.7 Suppose Y is a filter in a space \B−\ . For , the following are equivalent:
i) B is a cluster point of Y ii) cl B− Ö J ÀJ −Y × iii) there is a filter base UY for such that B is a cluster point of U iv) for every filter base UY for , B is a cluster point of U .
396 Proof It is clear from the definition and the following remarks that i)ÍÍ ii) iii).
iii)ÊB iv ) Suppose is a cluster point of UU and www is another base for Y . If F−© UY then FªFw for some set F−UU because is a base for Y . But every neighborhood of B intersects FBFB, so every neighborhood of also intersects ww . Therefore is a cluster point of U .
iv)Êñ i) This follows since YY is a base for .
Example 3.8
1) Let Y‘œÖJ© ÀJ ªÒ!ß"Ó× . Each real number < has a neighborhood R for which RªÎ Ò!ß "Ó and therefore R ÂYYaY Þ So ªÎÄ< so Î<Þ If <−Ò!ß"Ó , then <−E© cl E for each E−YY ß so < is a cluster point of . If <  Ò!ß "Ó , then R œ‘Y Ò!ß "Ó is a neighborhood of < disjoint from Ò!ß "Ó − Þ Therefore < is not a cluster point of YY . So the set of cluster points of is precisely the interval Ò!ß "ÓÞ
2) Let Y have the usual topology. œÖE© À E is finite × is a filter (check! ). For each 8 −Y , we have J œ Ö8 "ß 8 #ß ÞÞÞ× − and 8  J œ cl J Þ Therefore 8 is not a cluster point of YY . And since has no cluster points, certainly Y does not converge.
3) Let YYYYaYa be a filter in a space \ÄBÄCߪ . If and then BC and ªÞ Since gÂY ßevery neighborhood of B must intersect every neighborhood of C . Therefore, if \ is Hausdorff, we must have BœC , that is, a filter in a Hausdorff space can have at most one limit . (In fact, \\ is Hausdorff iff every filter in Y in has at most one limit. We could prove this directly,here and now (try it!). But this fact follows later “for free” from Theorem 2.6 via the duality between nets and filters: see Corollary 4.5 below.)
4. The Relationship Between Nets and Filters
Nets and filters are dual to each other in a natural way it is possible to move back-and-forth between them. Although this back-and-forth process is not perfectly symmetric, it is still very useful because the process preserves limits and cluster points.
Definition 4.1 Let ÐߟÑAY be a directed set. For a net ÐBÑ- in \ , we define its associated filter ÐÐBÑÑor, the filter generated by - : :
th Let XœÖBÀ-..A.- − ß ×œ the - tail of the net, and let U œÖXÀ−- -A × . U is nonempty since A----Ágß and each X--- Ág since B −X Þ Moreover, if $" and $# ,
then X∩XªX--"# - $ . Therefore the collection of tails U-A œÖXÀ−× - is a filter base and the filter it generates is the associated filter of ÐB- ÑÞ
397 Definition 4.2 For a filter YY in \ÐÑÀ , we define its associated net or, the net generated by
Let AYœÖÐBßJÑÀB−J− × and define ÐBßJÑŸÐBßJww Ñ if J ªJ w . Then Ð A ß ŸÑ is directed. (Why?) ) The net 0 ÀAY Ä \ defined by 0ÐÐBß J ÑÑ œ B is the associated net of .
Notice that ÐߟÑAY is not a poset: for example if BÁB−J−w , then ÐBß J Ñ ÐBww ß J Ñ and ÐB ß J Ñ ÐBß J Ñ but ÐBß J Ñ Á ÐB w ß J Ñ .
If we begin with a filter YY , form its associated net, and then form its associated filter w , we are back to where we started:
w YYY{{ associated net ÐB- Ñ associated filter œ
w To see this: a base for YA is the collection of tails ÖXÐBßJ Ñ À ÐBß J Ñ − ×Þ But ww ww w ww XÐBßJ Ñ œ Ö0ÐB ß J Ñ À ÐB ß J Ñ ÐBß J Ñ œ ÖB À ÐB ß J Ñ ÐBß J Ñ× ww w and w , where w w w So the œÖB ÀB −J J ©J JßJ −YYY × œ ÖJ ÀJ ©J − ל Þ collection of tails is Y itself !
On the other hand, if we begin with a net ÐB- À-A − × , form its associated filter Y , and then form the net associated with Y , we do not return to the original net ÐB- Ñ À
ÐB- Ñ{{ associated filter YY net associated to Á ÐB- Ñ
First of all, the net associated with Y has for its directed set AYAw œÖÐBßJÑÀB−J− ×Á Þ But the problem runs even deeper than that:
Consider the net ÐB88 Ñ in ‘ with directed set , where B œ ! for all 8 . The collection of tails is UY‘œ ÖÖ!××, which generates the associated œ ÖE © À ! − E× (a trivial ultrafilter). In turn, YAYA generates a net whose directed set œ ÖÐBß EÑ À B − E − × and in , Ð!ß Ö!×Ñ is a maximal element. The directed set for the new net is not even order isomorphic to !
Nevertheless, the following theorem shows that this back-and-forth process between associated nets and associated filters is good enough to be very useful for topological purposes.
Theorem 4.3 In any space \ ,
1) BÐBÑB is a cluster point of a net - iff is a cluster point of the associated filter Y . 2) BB is a cluster point of a filter Y iff is a cluster point of the associated net ÐBÑ-
Proof 1) BÐBÑÐBÑ is a cluster point of -- iff is frequently in every neighborhood of B iff X-- ∩RÁg for every tail X and every R−aB iff BœÖXÀ−× is a cluster point of the filter base U-A- iff B is a cluster point of Y .
2) We use 1). Given a filter Y , consider its associated net ÐB- ÑÞ By 1), B is a cluster point of ww ÐB- Ñ iff B is a cluster point of its associated filter YYY . But œ Þ ñ
Notice : if we had somehow proved part 2) first, and then tried to use 2) to prove part 1), we would run into trouble we would end up looking at a net different from the one we started
398 with. The “asymmetry” in the back-and-forth process between nets and filters shows up here. .
Theorem 4.4 In any space \ ,
1) a net ÐB- Ñ Ä B iff its associated filter Y Ä B . 2) a filter Y ÄB iff its associated net ÐBÑÄBÞ-
Proof The proof is left as an exercise. As in Theorem 4.3, part 2) follows “for free” from 1) using the duality between nets and filters. ñ
Corollary 4.5 A space \\ is Hausdorff iff every filter Y has at most one limit in .
Proof The result follows immediately by duality: use Theorem 4.4 and Theorem 2.6. ñ
The following theorem shows how subnets and larger filters are related: subnets generate larger filters and vice-versa.
Theorem 4.6 Suppose YA is a filter in \0ÀÄ\ generated by some net . (This is not a restriction on Y because every filter is generated by a net: for example, by its associated net.)
1) Each subnet of 0ª generates a filter ZY 2) Each filter ZYª0 is generated by a subnet of .
th Proof 1) A base for Y-A is ÖF-- À − × , where F is the - tail of 0Þ Suppose 9A À Q Ä and that th 0‰9..9 ÀQÄ\ is a subnet of 0 . The filter base ÖG.. À −Q× , where G is the tail of 0‰ , generates a filter Z .
Let J−Y-A...9.- . Then JªF-0 for some 0 − . Pick !!! −Q so that Ê Ð Ñ Þ Then the subnet tail G.-!! œÖ0‰9. Ð ÑÀ . .! שF ©J , so J is in the filter Z generated by 0‰ 9 . Therefore YZ© .
2) Conversely, let ZYZ be a filter containing . We claim is generated by a subnet of 0 . Let >2 BGBG-- denote the -UZ-AZ tail of 0œÖ−−×À . We claim that ∩ : , is a base for
ww UYZY is clearly base for some filter and © (since GGBª∩- ). On the other w hand, each BGB--−©YZ , so each ∩ − ZY so © Z
Let QœÖÐ0Z-0A0-0 ,GB∩∩-0-- ÑÀ G − ß ß − ß and 0ÐœB− ) GB × . (For each K∩F there ww is at least one such 000 because GB∩∩∩---Ág ). Order Q by defining Ð , GB ÑŸÐ , G Bw Ñ if w w -- Ÿ and GB∩©∩-w GB- . It is easy to check that is transitive and reflexive. In fact, ÐQß Ÿ Ñ is a directed set:
ww w ww w If --- KœKK− , , we have BBB--ww©∩ w - . Since ∩ Z , we have ww ww ww ww ww KÁg∩−∩BB-ww , so there exists a 0- such that 0ÑœBK ( 0 0 ww - ww . Thus wwww ww ww (000 ,K−Qß∩∩∩BBB--ww ) and we have ( , K K ww ) ( ,- ) and ww ww w w (,00K K∩∩BB--ww ) (, w ).
Define 9A900ÀQÄ by ÐÐ ,GB∩ - ÑÑœ . Then 0‰ 9 is a subnet of 0 :
399 Suppose -A!!!!!−− 0œB− . Pick any GGB Z and any 0- such that ( 0 )0-!!! ∩ .
Let .0!!œÐ ,GB! ∩∩--! ) −QÞ If .0 œÐ , GB Ñ . ! in Q , then 9. Ð Ñœ 00 !! -.
The filter generated by the subnet 0œÐÑ−Q‰∩9 has the set of tails CGB a base. If .0 , , we .-! !!!! claim that the tail C.-!! =K∩F! . If this is true, we are done , since the sets of this form, as noted above, are a base for the filter Z .
GœÖ0‰ÐÑÀ .! 9. . .!!! ×Þ If . −Q and . œÐßK∩FÑ œÐ 0-- . 0 , GB! ∩ ! Ñß
then GG∩F- ©!! ∩F---! , so 0‰9. Ð Ñœ0ÐÑ−K∩F 0 © G ∩F ! .
Therefore GÞ.-!!©∩FG!
Conversely, if C−K! ∩F-0! , then Cœ0ÐÑœB00A0- for some − ß ! .
Then .0œÐ , K! ∩ B0. ) −Q and .. ! so Cœ0 ( 0 ) œ0‰ 9. Ð Ñ−G ! .
Therefore Gª.-!!G! ∩F Þñ
The remark following Definition 2.10 is relevant here. To prove part 2) of Theorem 4.6: we need the more generous definition of subnets to be sure that we have “enough” subnets to generate all possible filters containing Y . In particular, the subnet defined in proving part 2) is not a “subnet” using the more restrictive definition for subnet.
Theorem 4.7 A point B−\ is a cluster point of the filter YZY iff there exists a filter ª such that Z ÄB.
Proof ÐÊÑIf such a filter ZaZ exists, then B © Þ Therefore each neighborhood of B intersects each set in ZYY , and therefore, in particular, intersects each set in . Therefore B is a cluster point of .
ÐÉÑ If B is a cluster point of YU , then the set œÖR∩JÀR− aYB and J− × is a filter base that generates a filter ZY . For J− ß we have Jœ\∩J− UYZ ß so © ; and for each R−aUaZZBB ßwe have RœR∩\− ß so © and therefore ÄBÞñ
Corollary 4.8 BÐBÑ\ÐBÑÄBÞ is a cluster point of the net -- in iff there exists a subnet . ()This result was stated earlier, without proof, as Theorem 2.11.
Proof BÐBÑB is a cluster point of- iff is a cluster point of the associated filter Y iff there exists a filter ZYªÄB with Z iff ÐB- Ñ has a subnet converging to B . ñ
Example 4.9 Think about each of the following parallel statements about nets and filters. Which ones follow “by duality” from the others?
w 1) If Bœ+- for all -- ! ß then 1 ) If Y consists of all sets containing + ,
ÐB- Ñ Ä + then Y Ä +
400 w 2) ÐB- Ñ Ä + iff every subnet 2 ) If YZ Ä + iff Ä + for every filter ZY ª converges to +
w 3) If a subnet of ÐB- Ñ has cluster point +ß 3 ) If + is a cluster point of ZZY and ª ß then +ÐBÑ+Þ is a cluster point of - then is a cluster point of Y
5. Ultrafilters and Universal Nets
Nets and filters are objects that can be defined in any set \ . The same is true for ultrafilters and universal nets. No topology is needed unless we want to talk about convergence, cluster points and other ideas that involve “nearness.” So many results in this section are purely set-theoretic.
Definition 5.1 An ultrafilter in is called fixed (or trivial) if . is called free (or hhh\Ág nontrivial) if . h œg For exampleÐÑœÖE©\À:−E×see Example 3.4 ) the ultrafilter Y is fixed (trivial).
Theorem 5.2 For an ultrafilter h in \ , then the following are equivalent.
i) for some :−\ , h œÖE©\À:−E×
ii) for some h œÖ:× :−\ iii) is fixed that is, . hhÁg
Proof It is clear that i)ÊÊ ii) iii)
iii)ÊÖB×ÂB−\ i) Suppose i) is false. Thenh for every (why? ) so by Theorem 3.5, for every . Therefore . \ ÖB× −hh B © Ö\ ÖB× À B − \× œ g ñ
Example 5.3 If is a filter and , might not be an ultrafilter. YYY Ág
Let Y be the filter in generated by ÖF88 À 8 − ×ß where F œ Ö"× ∪ Ö5 À 5 8×Þ Then ∞ Since neither nor contains one of the sets , Y„„œ8œ" F8 œ Ö"×Þ œ Ö#ß %ß 'ß ÞÞÞ× © F 8 neither „„Y nor is in . Therefore Y is not an ultrafilter.
Y„∪Ö × is a filter base that generates a filter Yw strictly larger than Y . Similarly, Y„∪Ö × generates a filter YYYY„„ww strictly larger than . Then wÁ ww (because and cannot be in the same filter). So Y can be “enlarged” in at least two different ways.
Theorem 5.4 If YYhh is a filter in \© , then for some ultrafilter .
401 Proof Let cZZœÖ À is a filter and ZY ª × , ordered by inclusion. c is a nonempty poset since .. Let be a chain in . We claim that . Yc−ÖÀ−M× Zααα c Z − c Clearly, and is a filter: ZYααªà Z since each , and each set in is nonempty. ZZααÁg Ág Z α If Eß F −ZZ , then both Eß F are in a single filter . Therefore αα! E∩F−ZZ , so E∩F− . In addition, if G ªE , then G − ZZ , so G − Þ αα! αα!
Therefore the chain has an upper bound in . By Zorn's Lemma, contains a ÖÀ−M×Zααα Z c c maximal element h . ñ
Example 5.5 For each 8−h , there is a fixed (trivial) ultrafilter 8 œÖE© À8−E×Þ It is the ultrafilter for which By Theorem 5.2, the 's are the only fixed ultrafilters in . There hh88œÖ8×Þ are also nontrivial (free) ultrafilters in À
Let FœÖ5−88U À5 8×Þ The collection œFÀ8−× { is a filter base and UY generates a filter . By Theorem 5.4 there is an ultrafilter hYª , and ∞ h ©FœgÞ8œ" 8 hY„ is not the only ultrafilter containing . If and are the sets of even and odd natural numbers, then U„∪Ö × and U ∪Ö × are filter bases that generate filters Ywww and Y . Then there are ultrafilters hYwªªÞ−−∩œg w and h ww Y ww Since „hh w , ww and „ , we have www Since w and ww , we know that w and ww , so hÁ h Þ Uh © Uh © h œg h œg hhww and are free ultrafilters.
It is a nice exercise to prove that if there is only one ultrafilter h containing YYh , then œÀ that is, if YY is not an ultrafilter, then can always be enlarged to an ultrafilter in more than one way.
If Y is a filter in , then Y− cc Ð Ð ÑÑ , so there are at most l cc Ð Ð ÑÑlœ##-i! œ# filters in .
Comment without proof: There are exactly #-- different filters in . In fact, there are 2 different - ultrafilters in , and since contains only countably many fixed ultrafilters h8 , there are in fact # free ultrafilters !
Theorem 5.6 An ultrafilter h in a space \ converges to each of its cluster points.
Proof If BªÄBœ is a cluster point of hZhZZhh , then there is a filter such that . But since is an ultrafilter. ñ
Corollary 5.7 An ultrafilter h in a X\# space has at most one cluster point.
We now define the analogue of ultrafilters for nets.
Definition 5.8 A net ÐB- Ñ in \ is called universal net (or ultranet ) if for every E © \ , ÐB- Ñ is either eventually in E\E or eventually in .
402 For example, a net ÐB-- Ñ which is eventually constant, say B œ : for all -- ! ß is a universal net. It is referred to as a trivial universal net because its associated filter is the trivial ultrafilter ÖE © \ À : − E×.
Theorem 5.9 In any set \ ,
1) a net ÐB- Ñ is universal iff its associated filter is an ultrafilter, and 2) a filter Y is an ultrafilter iff its associated net is universal.
Proof 1) ÐB- Ñ is universal iff for every E © \ , ÐB- Ñ is eventually in E or \ E iff for every E©\ , E or \E contains a tail of ÐBÑ- iff for every E©\ , E or \E is in the associated filter iff the associated filter is an ultrafilter.
2) We use duality. For a given a filter Y , consider its associated net ÐB- Ñ . ww By a),ÐB- Ñ is universal iff its associated filter YYY is an ultrafilter. But œ . ñ
Corollary 5.10 In any space \ ,
1) a subnet of a universal net is universal 2) a universal net converges to each of its cluster points 3) every net has a universal subnetÞ
Proof 1) If ÐB-- Ñ is universal, then ÐB Ñ generates an ultrafilter h . By Theorem 4.6, each subnet ÐB-. Ñ generates a filter Zhªœ . So Zh . Because the filter associated to ÐBÑÐBÑ--.. is an ultrafilter, is universal.
2) If BÐBÑB is a cluster point of the universal net - , then is a cluster point of the associated ultrafilter hh . By Theorem 5.6 ÄB and therefore ÐBÑÄB- .
3) Let YhY be the associated filter for the net ÐB- Ñ and let be an ultrafilter containing . By
Theorem 4.6(2), h is generated by a subnet ÐB--. Ñ of ÐB Ñ . Since the filter associated with ÐB -. Ñ is an ultrafilter, ÐB-. Ñ is a universal net. ñ
Corollary 5.11 A universal net in a X# space has at most one cluster point.
Both nets and filters are sufficient to describe the topology in any space, so we should be able to use them to describe continuous functions.
Theorem 5.12 Suppose \] and are topological spaces, 0À\Ä]+−\ and . The following are equivalent:
1) 0+ is continuous at 2) whenever a net ÐB-- Ñ Ä + in \ , then Ð0ÐB ÑÑ Ä 0Ð+Ñ in ] 3) whenever a universal net ÐB-- Ñ Ä + in \ , then Ð0ÐB ÑÑ Ä 0Ð+Ñ in ] 4) whenever +ÐBÑ\0Ð+ÑÐ0ÐBÑÑ is a cluster point of a net -- in , then is a cluster point of
403 in ] 5) whenever a filter base UUUÄ+in \ , then the filter base 0Ò ÓœÖ0ÒFÓÀF− ×Ä0Ð+Ñ in ] 6) whenever an ultrafilter hhÄ+in \ , then the filter base 0Ò ÓÄ0Ð+Ñ in ]Þ 7) whenever +\0Ð+Ñ is a cluster point of a filter base U in , then is a cluster point of the filter base 0ÒU Ó in ]Þ
Proof 1)ÊZ 2) Suppose is a neighborhood of 0Ð+Ñ0 . Since is continuous at + , there is a neighborhood Y of + such that 0ÒY Ó © Z Þ If ÐB-- Ñ Ä + , then ÐB Ñ is eventually in Y so Ð0ÐB - ÑÑ is eventually in Z Þ Therefore Ð0ÐB- ÑÑ Ä 0Ð+ÑÞ
2)Ê 3) This is immediate.
3)Ê+ 4) If is a cluster point of ÐBÑ--- , then there is a subnet ÐBÑÄ+ÞÐBÑ.. Let / be a universal subnet of ÐB--.. Ñ . Then ÐB// Ñ Ä + and by iii), Ð0ÐB - . ÑÑ Ä 0Ð+ÑÞ Then Ð0ÐB - ÑÑ has a subnet converging to 0Ð+Ñß so 0Ð+Ñ is a cluster point of Ð0ÐB- ÑÑÞ
4)Ê0 1) Suppose is not continuous at + . Then there is a neighborhood Z0Ð+Ñ of such that 0ÒRÓ ©ÎZ for every R−aAa++ Þ Let œ , ordered by reverse inclusion, and define a net 1À A Ä\ by1ÐRÑœBœRRRR a point in R for which 0ÐBÑÂZÞ Since ÐBÑÄ+ÐBÑ , has a cluster point at + . But the net Ð0ÐBR ÑÑ does not have a cluster point at 0Ð+Ñ because Ð0ÐBR ÑÑ is never in Z . Therefore 4) fails.
Knowing that 1)-4) are equivalent, we could show that each of 2)-4) is equivalent to its filter counterpartÍ e.g., that 2) 5), etc. This involves a little more than simply saying “by duality” because, in each case, the function 0 also comes into the argument. Instead, for practice, we will show directly that 1)ÊÊÊÊ 5) 6) 7) 1).
It is easy to check that if UUU is a filter base in \ , then 0Ò ÓœÖ0ÒFÓÀF− × is a filter base in ] .
1)Ê 5) Suppose UUÄ + in \ and let 0ÒFÓ − 0Ò ÓÞ If R is any neighborhood of 0Ð+Ñ in ] , then by continuity 0ÒRÓ" is a neighborhood of + in \ . Therefore 0ÒRÓªF " for some F−U , so R ª 0Ò0" ÒRÓÓ ª 0ÒFÓÞ Therefore 0ÐU Ñ Ä 0Ð+ÑÞ
5)Ê 6) This is immediate.
6)Ê+ 7) Suppose is a cluster point of the filter base UhU . We can choose an ultrafilter ª with hhÄ + . By 6) , 0Ò Ó Ä 0Ð+Ñ . Therefore the filter hw generated by 0Ò h Ó converges to 0Ð+Ñ , so ww haª0Ð+Ñ. Since 0ÒÓ© Uh , each set in 0ÒÓ U intersects every set in a0Ð+Ñ . Therefore 0Ð+Ñ is a cluster point of 0ÒU Ó .
7)Ê0 1) Suppose is not continuous at + . Then there is a neighborhood Z0Ð+Ñ of such that " 0ÒRÓ ©ÎZ for all R−aa++ , that is, R0 ÒZÓÁg for all R− Þ The collection " UaœÖR0 ÒZÓÀR−+ ×is a filter base (why? ) that has a cluster point at + . However 0Ò U Ó does not cluster at 0Ð+Ñ since no set in 0ÒU Ó intersects Z . ñ
Corollary 5.13 Let be a net in the product . Then in iff ÐB-α- Ñ \ œ Ö\ Àα − E× ÐB Ñ Ä B \ (11α-ÐB ÑÑ Ä α ÐBÑ in \ α for each α − E .
404 Proof If ÐB-α-αα Ñ Ä B − \ , then by continuity Ð11 ÐB ÑÑ Ä ÐBÑ − \ for every α − E .
Conversely, suppose Ð11α- ÐB ÑÑ Ä α ÐBÑ for every α and let Y œ Y α"# ß Y α ß ÞÞÞß Y α 8 be a basic open set containing B in \ . For each 3 œ "ß ÞÞÞß 8 we have Ð11α-333 ÐB ÑÑ Ä α ÐBÑ − Y α . Therefore we ‡‡ can choose a -A33"8−ÐBÑ−Y so that 1α-33 α when -- . Pick - - , ... , - . If -- , we have ‡ 1--α-33ÐB Ñ − Y α for each 3 œ "ß ÞÞÞß 8 . Therefore B - − Y for so ÐB - Ñ Ä BÞ ñ
6. Compactness Revisited and The Tychonoff Product Theorem
With nets and filters available, we can give a nice characterization of compact spaces in terms of convergence.
Theorem 6.1 For any space \ , the following are equivalent:
1) \ is compact 2) if is a family of closed sets with the finite intersection property, then YY Ág 3) every filter Y has a cluster point 4) every filter can be enlarged to a filter that converges 5) every net has a cluster point 6) every net has a convergent subnet 7) every universal net converges 8) every ultrafilter converges.
Proof We proved earlier that 1) and 2) are equivalent (see Theorem IV.8.4 ).
2) Ê\ 3) If YY is a filter in , then has the finite intersection property, so ÖJÀJ−× cl Y is a family of closed sets also with the finite intersection property. By ii), cl so is a bB− Ö JÀJ−Y × B cluster point of Y .
3)ÊBª 4) If YY is a filter, then has a cluster point so, by Theorem 4.7, there is a filter ZY such that Z ÄB.
4)ÊÐBÑ 5) If - is a net, consider the associated filter YZY . By 4) there is a filter ª where
ZZÄB−\. is generated by a subnet ÐB--.. Ñ and by duality, ÐB ÑÄB .
5)ÊÐBÑ 6) If - has a cluster point B , then by Corollary 4.8, there is a subnet ÐBÑÄBÞ-.
6)ÊÐBÑ 7) If -- is a universal net, then 6) gives that ÐBÑ has a subnet that converges to a point B . Then B is a cluster point of ÐBÑÞ-- Since ÐBÑ is universal, ÐBÑÄB - by Corollary 5.10.
7)Ê 8) This is immediate from the duality between universal nets and ultrafilters (Theorems 5.9 and 4.4)
405 8)Ê\ 1) Suppose is not compact and let hα œÖYÀ−E×α be an open cover with no finite subcover. Then for any αα"# , ,..., α 8−E\ÁY∪Y∪ÞÞÞ∪Y , αα"# α 8 ; so, by complements, gÁÐ\Yαα"8 Ñ∩ÞÞÞ∩Ð\Y Ñ. Therefore fα œÖ\Y α À −E× is a collection of closed sets with the finite intersection property. The set of all finite intersections of sets from f is a filter base which generates a filter YiY , and we can find an ultrafilter ª . Every point BY is in ααα for some α ßBÂ\YœÐ\YÑÞB so cl Therefore is not a cluster point of ii . In particular, this implies ÄÎB . Since B was arbitrary, this means i does not converge. ñ
Theorem 6.1 gives us a fresh look at the relationship between some of the “compactness-like” properties that we defined in Chapter IV:
\\ is compact iff is sequentially compact iff every net has a convergent subnet every sequence has a convergent subsequence (Theorem 6.1) (Definition IV.8.7)
àá
\ is countably compact iff every sequence has a cluster point (Theorem IV.8.10) iff every sequence has a convergent subnet (Corollary 4.8)
It is now easy to prove that every product of compact spaces is compact.
Theorem 6.2 (Tychonoff Product Theorem) Suppose is compact iff \œ Ö\Àα α −E×ÁgÞ \ each \α is compact.
Proof For each α1 , \œαα Ò\Ó so if \ is compact, then each \ α is compact.
Conversely, suppose each \ÐBÑ\œÖ\À−E×α- is compact and let be a universal net in αα . " For each α1 , ÐÐBÑÑα- is a universal net in \ α (Check: ifE©\α- , then ÐBÑ is eventually in 1α ÒEÓ or " 1α Ò\α EÓ, so ÐÐBÑÑ1α- eventually in E or \ EÞ ) But \α is compact, so by Theorem 6.1 Ð1α- ÐBÑÑÄsome point D α −\ α Þ Let DœÐDÑ−\ α . By Corollary 5.13, ÐBÑÄD - . Since every universal net in \\ converges, is compact by Theorem 6.1. ñ
Remark: A quite different approach to the Tychonoff Product Theorem is to show first that a space \ is compact iff every open cover by sub basic open sets has a finite subcover. This is called the Alexander Subbase Theorem and the proof is nontrivial: it involves an argument using Zorn's Lemma or one of its equivalents. After that, it is fairly straightforward to show that any cover of \œ Ö\Àα α −E× by sets of " the form 1α ÒYα Ó has a finite subcover. See Exercise E10.
406 At this point we restate a result which we stated earlier but without a complete proof (Corollary VII.3.16).
Corollary 6.3 A space \ is Tychonoff iff it is homeomorphic to a subspace of a compact Hausdorff space. (In other words, the Tychonoff spaces are exactly the subspaces of compact Hausdorff spaces.)
Proof A compact Hausdorff space is X% , and therefore Tychonoff. Since the Tychonoff property is hereditary, every subspace of a compact X# space is Tychonoff.
Conversely, every Tychonoff space \ is homeomorphic to a subspace of some cube Ò!ß "Ó7 . This cube is Hausdorff and it is compact by the Tychonoff Product Theorem. ñ
Remark Suppose \ is embedded in some cube Ò!ß "Ó77 Þ To simplify notation, assume \ © Ò!ß "Ó . 7 Then \© cl \œO©Ò!ß"ÓÞ O is a compact X# space containing \ as a dense subspace and O is called a compactification of \ . Since every Tychonoff space can be embedded in a cube, we have therefore shown that every Tychonoff space \ has a compactification.
Conversely, if O\O\ is a compactification of , then is Tychonoff and its subspace is also Tychonoff. Therefore \ has a compactification iff \ is Tychonoff .
Our proof of the Tychonoff Product Theorem used the Axiom of Choice (AC) in the form of Zorn's Lemma (to get the necessary universal nets or ultrafilters). The following theorem shows that, in fact, the Tychonoff Product Theorem and AC are equivalent. This is perhaps somewhat surprising since AC is a purely set theoretical statement while Tychonoff's Theorem is topological. On the other hand, if “all mathematics can be embedded in set theory” then every mathematical statement is purely set theoretical.
Theorem 6.5 (Kelley, 1950) The Tychonoff Product Theorem implies the Axiom of Choice (so the two are equivalent).
Proof Suppose Ö\α Àα − E× is a collection of nonempty sets. The Axiom of Choice is equivalent to the statement that (see Theorem 6.2.2 ). α−E\Ágα Let where and give the very simple topology ]αα œ \ ∪ Ö:× : Âα−E \ α ] αg α œ Ö] α ß Ö:×ß g×Þ Then is compact, so is compact by the Tychonoff Product Theorem. ]]œ]ααα−E
" Ö:× is open in ]αα , so \ is closed in ] α . Therefore Y1 œ Öα Ò\ α Ó À α − E× is a family of closed sets in ] . We claim that Y has the finite intersection property.
Suppose αα"#ß ß ÞÞÞß α 8 − E . Since the \ααα 's are nonempty, there exist points B"" − \ ß ÞÞÞß
B−\αα88.
Define by: 0ÀEÄα−E ]α Bœif αα for αα−E, 0Ð Ñœ α3 3 :Áif αααα"#, ,..., 8
To be more formal since this is the crucial set-theoretic issue in the argument we can formally and precisely define 0 in ZF by :
407 0 œÖÐααααα ßCÑ−E‚ ] ÀÐ œ •CœB Ñ”ÞÞޔРœ •CœB Ñ α−E αα"8"8 α ” ÐÐαα Á" Ñ•ÞÞÞ•Ð αα Á8 ÑÑ•Cœ:Ñ×
Then " " 0 −11αα"8 Ò\αα"8 Ó ∩ ÞÞÞ ∩ Ò\ ÓÞ
Since has the finite intersection property and is compact, . If , then YYY]Ág1− and therefore . 1−αα−E \αα −E \ Ág ñ
If and any 's are noncompact, then is noncompact. And we note that if infinitely \Ágαα \ \ α many 's are noncompact, then is “dramatically” noncompact as the following theorem \\αα indicates.
Theorem 6.6 Let . If infinitely many of the 's are not compact, then \œ Ö\αα Àα −E×Ág \ every compact closed subset of \ is nowhere dense. (Thus, all closed compact subsets of \ are “very skinny” and “far from” being all of \ .)
Proof Suppose F\F is a compact closed set in and that is not nowhere dense. Then there is a point
B and indices αα"# , ß ÞÞÞß α 8 such that B − Yαα"# ß Y ß ÞÞÞ Y α 8 © int Ð cl FÑ œ int F © FÞ Then 1ααÒFÓ œ \ for α Á αα"# , ß ÞÞÞß α 8 , so \ α is compact if α Á αα "# , ß ÞÞÞß α 8 . ñ
7. Applications of the Tychonoff Theorem
We have already used the Tychonoff Theorem in several ways (see, for example, Corollary 6.3 and the remarks following.) It's a result that is useful in nearly all parts of analysis and topology, although its full generality is not always necessary. In this section we sketch how it can be used in more “unexpected” settings. The following examples also provide additional insight into the significance of compactness.
The Compactness Theorem for Propositional Calculus
Propositional calculus is a part of mathematical logic that deals with expressions such as :•; , :”; , :Ê;, µ: , Ð:Ê;Ñ”< , etc. Letters such as :ß;ß<ßÞÞÞ are often used to represent “propositions” that can have “truth values” T (true) or F (false). These letters are the “alphabet” for propositional calculus. For example, we could think of :##œ& as representing the (false) proposition “ ” or the (true) proposition “aB −‘‘ ÐB ! Ê bC − ÐC# œ BÑÑ ”. However, : could not represent an expression like “Bœ& ”, because this expression has no truth value: it contains a “free variable” B .
In propositional calculus, propositions :ß ;ß ÞÞÞ are thought of as “atoms” that is, the internal structure of the propositions :ß ;ß ÞÞÞ (such as variables and quantifiers) is ignored. Propositional calculus deals with “basic” or “atomic” propositions such as :ß ;ß ÞÞÞ , with compounds built up from them such as Ð: ” ;Ñ and µ Ð µ : ” ;Ñ , and with the relations between their truth values. We want to allow the
408 possibility of infinitely many propositions, so we will use EE"8 , ... , , ... as our alphabet instead of the letters :ß ;ß ÞÞÞ that one usually sees in beginning treatments of propositional calculus.
Here is a slightly more formal description of propositional calculus.
Propositional calculus has an alphabet TTœ ÖE"# ß E ß ÞÞÞß E 8 ß ÞÞÞ× . We will assume is countable, although that restriction is not really necessary for anything we do. Propositional calculus also has connective symbols: ( , ) ,”µÞ , and
Inductively we define a collection j of well-formed formulas (called wffs , for short) that are the “legal expressions” in propositional calculus:
1) for each 8E , 8 is a wff 2) if 9< and are wffs, so are Ð”Ñ 9< and µ 9 .
For example, ÐE ("#$ ”E ) ”µEÑ % and µÐE”EÑ "% are wffs but the string of symbols ( ” A $ ”” ) is not a wff. If we like, we can add additional connectives •Ê and to our propositional calculus ß defining them as follows:
Given wffs 9< and ,
9<•œµÐµ”µÑ 9 < and 9<<Êœ”µ 9 .
Since •Ê and can be defined in terms of ” and µ , it is simpler and involves no loss to develop the theory using just the smaller set of connectives.
A truth assignment is a function =EÞ that assigns a truth value to each 8 More formally, =ÀT ÄÖXßJ×, so =−ÖXßJ×i! œGÞ We give G the product topology. By the Tychonoff Theorem, GG is compact (in fact, by Theorem VI.2.19, is homeomorphic to the Cantor set).
A truth assignment = can be used to assign a unique truth value to every wff in j , that is, we can extend ==ÀÄÖXßJ× to a function j as follows:
For any wff 5 we define:
if 555œE8 , then =Ð Ñœ=ÐÑ
X=ÑœX=ÐÑœXif (9< or if 59<œ” , then =( 5 Ñœ J otherwise X=ÑœJif (9 if 59œµ , then =ÐÑœ 5 J=ÑœXif (9
We say that a truth assignment ==ÐÑœX satisfies a wff 55 if . A set of wffs D is called satisfiable if there exists a truth assignment =−ÖXßJ×i! such that = satisfies 55D for every − .
For example, D œÖEßE”EßE×"" ## is satisfiable ( We can use any = for which =ÐE"# Ñ œ =ÐE Ñ œ X Ñ
409 D œÖEß "" µ ( E ” µE # ) × is not satisfiable ( If =ÐE" Ñ œ X , then =µE”µEÑœJ(())"#
Theorem 7.1 (Compactness Theorem for Propositional Calculus) Let D be a set of wffs. If every finite subfamily of DD is satisfiable, then is satisfiable.
Proof Let GœÖXßJ×i! and suppose that every finite subset of D is satisfiable. Then for each 5D−, E55 œÖ=−GÀ= satisfies 5 ×Ág . We claim that E is closed in G :
Suppose =ÂE5 . We need to produce an open set Y containing = for which Y∩E5 œgÞ
For a sufficiently large 8 , the list E"8 ß ÞÞÞß E will contain all the letters that occur in 5 . i! Let Y œ Ö> − GÀ >33 œ = , 3 œ "ß ÞÞÞß 8× œ Ö= " × ‚ ÞÞÞ ‚ Ö= 8 × ‚ ÖX ß J × . In other words, Y is the set of truth assignments that agree with = for all the letters E"8 ß ÞÞÞß E that may occur in 55 . Since = fails to satisfy , each >−Y also fails to satisfy 55 , so −Y ©GE5 .
The E5 's have the finite intersection property in fact, this is precisely equivalent to saying that every finite subset of is satisfiable. Since is compact, , i.e., is satisfiable. D5DDGÖEÀ−×Ágñ 5
410 If we assume the Alexander Subbase Theorem (see the remarks following Theorem 6.2, as well as Exercise E10 ), then we can also prove that the Compactness Theorem 7.1 is equivalent to the statement that GœÖXßJ×i! is compact.
Suppose the Compactness Theorem 7.1 is true. To show that G is compact it is sufficient, by the Alexander Subbase Theorem, to show that every cover of G by sub basic open sets has a " " finite subcover. Each subbasic set has the form 1188ÒÖX ×Ó or ÒÖJ ×Ó .
Taking complements, we see that it is sufficient to show that:
if Y is any family of closed sets with the finite intersection property and each set in Y has form " [{ " [ or " [ " [ , then . G 1188 X ×Ó œ ÖJ ×Ó G 11 88 ÖJ ×Ó œ ÖX ×Ó Y Á g
Let YαœÖJα À −E× be any such family. For each α −E , define a wff
" EJœÖX×Ó8 if α 18 [ 5α œ " µ E8 if Jα œ18 [ ÖJ ×Ó
Clearly,5αα is satisfied precisely by the truth assignments in J .
Let D5αœÖαα À −E× . Since the J 's have the finite intersection property, any finite subset of is satisfiable. By the Compactness Theorem, is satisfiable, so . DDY Ág ì
Note: If the propositional calculus is allowed to have an uncountable alphabet of cardinality 7 , then the compactness theorem is equivalent to the statement that ÖX ß J ×7 is compact; the proof requires only minor notational changes.
A “map-coloring" theorem
Imagine that Q is a (geographical) map containing infinitely many countries G"# , G ß ÞÞÞß G 8 ß ÞÞÞ . A valid coloring - of Q with 4 colors (red, white, blue, and green, say) is a function
- À ÖG"# ß G ß ÞÞÞß G 8 ß ÞÞÞ× Ä ÖVß [ ß Fß K× such that no two adjacent countries are assigned the same color.
Intuitively, if Q doesn't have a valid covering, it must be because some “finite piece” of the map Q has a configuration of countries for which a valid coloring can't be done. That is the content of the following theorem.
Theorem 7.2 Suppose Q is a (geographical) map with infinitely countries G"8 ß ÞÞÞß G ß ÞÞÞ . If every finite submap of QQ has a valid coloring, then has a valid coloring. (Any reasonable definition of “adjacent” and “submap” will work in the proof.)
Proof Consider set of all colorings G œÖVß[ßFßK×i! with the product topology. For each finite submap JZœÖ-−GÀ- , let J is a valid coloring of J×ÁgÞ We claim that ZJ is closed in GÞ
411 Suppose -ÂZJ . We need to produce an open set Y with -−Y©GZJ Þ If 8 is large enough, the list G"8 ß ÞÞÞG will include all the countries in the submap J . Then the open set
i! Y œ Ö-ÐGÑׂÞÞÞ‚Ö-ÐG" 8 ÑׂÖVß[ßFßK×
worksY any coloring in is invalid because it colors the countries in J- the same way does.
If JJ" and # are finite submaps of Q , then so is J∪JJ∪J "# . Since "# has a valid coloring by hypothesis, Z©Z∩ZÁgÞJ∪J"# J " J # Therefore Y œÖZÀJ J a finite submap of Q× has the finite intersection property. Since is compact, , and any is a valid covering of the whole GÁg-−YY map Qñ .
( It is clear that a nearly identical proof would work for any finite number of colors and for maps with uncountably many countries.)
412 Exercises
E1. Let \+−\ÞR−ßB−RÞ be a topological space and For each aa+R pick a point If we order + by reverse inclusion, then ÐBRR Ñ is a net in \ . Prove that ÐB Ñ Ä +Þ
E2. a) Let ÐGß Ÿ Ñ be an uncountable chain in which each element has only countably many predecessors. Suppose 0ÀGÄ‘- and that 0ÐÑ! for each - −G . Show that the net 0 does not converge to ! in ‘ .
b) Give an example to show that part a) is false if ÐGß Ÿ Ñ is an uncountable poset in which each element has only countably many predecessors.
b) Is it possible to have a net 0ÀÒ!ß=‘# ÑÄ and that 0ÐÑ! - for each - ? Is it possible that 0 converges to ! in ‘ ? (Recall that =αα denotes the first ordinal with i predecessors .)
E3. Suppose \ПÑ− is a compact Hausdorff space and that A-A , is a directed set. For each , let
EE©EŸ- be a nonempty closed subset in \ such that --#" iff --"# . Prove that ÖE- À-A − × Á gÞ
E4. a) Let 0ÀA-αA Ä\ be a net in a space \ and write 0ÐÑœBÞ- For each − , let th XœÖBÀα--α } œ “the α tail of the net.” Show that a point B−\ is a cluster point of ÐBÑ- iff cl B− Ö Xα ÀαA − ×Þ b) Suppose is a cluster point of the net a product { . Show that for each , BÐBÑ\À−E×-α αα 11ααÐBÑ − \ is a cluster point of the net ( α- ÐB ÑÑÞ
c) Give an example to show that the converse to part b) is false.
d) Let Ð\ß .Ñ be a metric space and 0 À Ò!ß=α" Ñ Ä \ a function given by 0Ð Ñ œ Bα . Show that the net ÐBαα Ñ converges iff ÐB Ñ is eventually constant.
E5. a) Suppose \ is infinite set with the cofinite topology. Let Y be the filter generated by the filter base consisting of all cofinite sets. To what points does Y converge? b) Translate the work in part a) into statements about nets.
E6. Show that if a filter YhYh is contained in a unique ultrafilter , then œ . (Thus, if Y is not an ultrafilter, it can enlarged to an ultrafilter in more than one way.)
E7. a) State and prove a theorem of the form: Suppose B\ is a point in a space . Then aB is an ultrafilter Í ....
b) Prove or disprove: Suppose \Ág and that Y is a maximal family of subsets of \ with the finite intersection property. Then Y is an ultrafilter.
413 E8. a) Let YY be a filter in a set \ . Prove that is the intersection of all ultrafilters containing Y .
b) Let hh be an ultrafilter in \ and suppose that E"8 ∪ ÞÞÞ ∪ E − . Prove that at least one E 3 must be in h . (This is the filter analogue for a fact in ring theory: in a commutative ring with a unit, every maximal ideal is a prime ideal.)
c) Give an example to show that part b) is not true for infinite unions.
d) “By duality,” there is a result similar to b) about universal nets. State the result and prove it directly.
E9. Let hD55gD be a free ultrafilter in and let œ∪Ö× , where  . Define a topology on by gœÖSÀS© or SœY∪Ö 5h × where Y− ×
a) Prove that DD is T% and that is dense in .
b) Prove that a free ultrafilter h on cannot have a countable base. Hint: Since hh is free, each set in must be infinite. Why?
c) Prove that no sequence in 5 can converge to (and therefore there can be no countable neighborhood base at 5D in ) (Hint: Work you did in b) might help.)
Thus, D is another example of a countable space where only one point is not isolated and which is not first countable, See the space P in Example III.9.8.
d) How would the space Dh be different if were a fixed ultrafilter?
Note: if hhhDDww is a free ultrafilter on and Á , then the corresponding spaces w and may not be homeomorphic: the neighborhood systems of 5 may look quite different. In this sense, free ultrafilters in do not all “look alike.”
E10. Suppose Ð\ßgg Ñ has a some property T . is called a maximal - T topology (or minimal - T topology) if any larger (smaller) topology on \TÐ\ßÑ fails to have property . Prove that if g is a compact Hausdorff space, then g is maximal-compact and minimal-Hausdorff. (Compare Exercise IV E23.) In one sense, this “justifies” the choice of the product topology over the box topology: for a product of compact Hausdorff spaces, a larger topology would not be compact and a smaller one would not be Hausdorff. The product topology is “just right” to ensure that the property “compact Hausdorff” is productive.
E11. Show that the map coloring Theorem 7.2 is equivalent to the statement that ÖVß[ßFßK×i! is compact.
414 E12. A family UU of subsets of \\ is called inadequate if it does not cover , and is called finitely inadequate if no finite subfamily covers \ .
a) Use Zorn's Lemma to prove that any finitely inadequate family U is contained in a maximal finitely inadequate family.
b) Prove (by contradiction) that a maximal finitely inadequate family U has the following property: if G"8 ß ÞÞÞß G are subsets of \ and G " ∩ ÞÞÞ ∩ G 8 −UU ß then at least one set G 3 − Þ
c) The following are equivalent (Alexander's Subbase Theorem)
i) \\ has a subbase ff such that each cover of by members of has a finite subcover
ii) \ has a subbaseff such that each finitely inadequate collection from is inadequate
iii) every finitely inadequate family of open sets in \ is inadequate
iv) \ is compact
d) Use c) to prove the Tychonoff Product Theorem.
E13. This exercise gives still another proof of the Tychonoff Product Theorem. Suppose \α is compact for all . We want to prove that is compact. We proceed assuming is not α −E \α \ compact.
a) Show that there is a maximal open cover h of \ having no finite subcover.
b) Show that if S is open in \ and S Âhh , then there are sets Y"8 ß ÞÞÞß Y − such that ÖY"8 ß ÞÞÞß Y ß S×covers \.
c) Show that for each αh , ÖZααα ©\ À Z is open and Z α − × cannot cover \ α Þ Conclude that for each α we can choose B−\αα so that BÂ α any open set Z α for which Zα −h Þ
d) Let BœÐBÑ−\ααα and suppose B−Y −h Þ Pick open sets Z33 ©\ so that
B − Zαα" ß ÞÞÞß Z5 © Y Þ Explain why each Z α3 Â h .
e) Show that for each 3 œ "ß ÞÞÞß 5 , there is a finite family hh33 © such that h ∪ Zα3 covers \.
f) Show that 5 h ∪ Ö Z ß ÞÞÞß Z × covers \ , and then arrive at the contradiction that 3œ" 3 " α5 5 covers . 3œ"h ∪ÖY× \
415 Chapter IX Review
Explain why each statement is true, or provide a counterexample.
1. Order GÐ\Ñ by 0 Ÿ 1 iff 0ÐBÑ Ÿ 1ÐBÑ (in ‘ ) for every B − \ . Then ÐGÐ\Ñß Ÿ Ñ is a directed set.
2. BE\ is a limit point of in the space iff there exists a filter YYY such that EÖB×−ÄB and .
3. A set Y\Y is open in iff belongs to every filter Y which converges to a point of Y .
4. In a space , let is a filter converging to . Then . \FYYB œÖÀ B× ÖÀ YYFa −BB ל
5. Suppose TT is a family of subsets of a space \EßF−E∩FªG such that if then for some G−TT. Suppose ÐBÑ- is a net which is frequently in each set of . Then ÐBÑ- has a subnet which is eventually in each set in T .
6. If a net ÐB-- Ñ Ä B in \ and l\l œ i! , then ÐB Ñ has a subsequence (i.e., a subnet whose directed set is ) which converges to B .
7. If \l\lœ8 is a nonempty finite set and , then there are exactly 28 1 different filters and exactly 8\ different ultrafilters on .
8. A universal sequence must be eventually constant.
9. Suppose \œÖ\l\l× is infinite. The collection f A ©\À | E | is an ultrafilter.
10. In ‘Y , a filter ÄB iff a % ! bJ− Y such that B− F and diam ÐJÑ % .
11. If \\ is compact, then every net in has a convergent subsequence. (Note: a “subsequence of a net” is a subnet whose directed set is .)
12. If BÁC in a space \ and if aaBC ∪ generates a filter, then \ is not X " .
13. If Y= is a filter in Ò!ß" Ó , then there must be a filter ZYZ ª such that has a cluster point.
416 14. Call a set in Ò!ß "Ó an endset if it has the form Ò!ß%% Ñ or Ð" ß "Ó for some % ! . To show Ò!ß "Ó is compact, it is sufficient to show that any cover by endsets has a finite subcover.
15. Every separable metric space Ð\ß .Ñ has an equivalent totally bounded metric.
16. Suppose A is the collection of finite subsets of Ò!ß "Ó , directed by © , and 0ÐJ Ñ œ " for all J − A . The net 0"Ò!ß"Ó converges to in .
417