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Chapter X Compactifications

1. Basic Definitions and Examples

Definition 1.1 Suppose 2À\Ä] is a of \ into ] , where ] is a compact X# . If 2Ò\Ó is dense in ] , then the pair Ð] ß 2Ñ is called a compactification of \ .

By definition, only Hausdorff spaces \ can (possibly) have a compactification.

If we are just working with a single compactification ]\ of , then we can usually just assume that \©] and that 2 is the identity map  so that the compactification is just a compact that contains \\ª] as a dense subspace. In fact, if , we will always assume that 2 is the identity map unless something else is stated. We made similar assumptions in discussing of the completion of a space Ð\ß .Ñ in Chapter IV.

However, we will sometimes want to compare different compactifications of \ (in a sense to be discussed later) and then we may need to know how \] is embedded in . We will see that different dense 2\ of into the same space ] can produce “nonequivalent” compactifications. Therefore, strictly speaking, a “compactification of \Ð]ß2Ñ ” is the pair .

If, in Definition 1.1, \2Ò\Ó] is already a compact Hausdorff space, then is closed and dense in and therefore 2Ò\Ó œ ] . Therefore, topologically, the only possible compactification of \ is \ itself.

The next theorem restates exactly which spaces have compactifications.

Theorem 1.2 A space \ has a compactification iff it is a .

Proof See the remarks following Corollary IX.6.3. ñ

Example 1.3 The W2" can be viewed as a compactification of the real , ‘ . Let be the “inverse projection” pictured below: here 2Ò‘‘ Ó œ W"  Ö North Pole × . We can think of 2Ò Ó as a “bent” topological copy of ‘ , and the compactification is created by “tying together” the two ends of ‘ by adding one new “ at ” (the North Pole).

418 Since lW""  2Ò‘‘ Ól œ " , ÐW ß 2Ñ is called a one-point compactification of . (We will see in Example 4.2 that we can call WÞ" the one-point compactification of ‘ )

Example 1.4

1) Ò  "ß "Ó is a compact Hausdorff space containing Ð  "ß "Ñ are a dense subspace, so Ò  "ß "Ó is a “two-point” compactification of Ð  "ß "Ñ (with 2 œ 3Ñ .

2) If 2 À‘‘ Ä Ð  "ß "Ñ is a homeomorphism, then 3 ‰ 2 À Ä Ò  "ß "Ó gives a “two- point” compactification of ‘ . It is true (but not so easy to prove) that there is no 8 -point compactification of ‘= for #8! Þ

Example 1.5 Suppose ]œ\∪Ö:× is a one-point compactification of \ . If S is an open containing :] in , then Oœ]S©\O and is compact. Therefore the open sets containing :\ are the complements of compact of . (Look at open neighborhoods of the North Pole :W in the one-point compactification " of ‘ ; a for the open neighborhoods of : consists of complements of the closed (compact!) arcs that do not contain the North Pole.) Suppose B−Yß where Y is open in \ . Because ] is Hausdorff, we can find disjoint open sets Z and [ in ] with B−Z and :−[Þ Since :ÂZ , we have that Z ©\ and therefore Z ∩\œZ is also open in \ . Since B−Y ∩Z , we can use the regularity of \ to choose an K in \ for which B−K© cl\ K©Y∩Z ©YÞ But K©Z ©] [ (a in ] Ñß so cl] K©] [ ©\ . Therefore cl\]]\Kœ\∩ cl Kœ cl K , so cl K is also closed in ]Þ So cl \ K is a compact neighborhood of BY inside . This shows that each point B−\ has a neighborhood base in \ consisting of compact neighborhoods.

The property in the last sentence is important enough to deserve a name: such spaces are called locally compact.

419 2. Local Compactness

Definition 2.1 A Hausdorff space \B−\ is called locally compact if each point has a neighborhood base consisting of compact neighborhoods Þ

Example 2.2

1) A is locally compact.

2) ‘8 is locally compact: at each pointBB , the collection of closed balls centered at is a base of compact neighborhoods. On the other hand, neither  nor is locally compact. (Why? Ñ

3) If \\ is a compact Hausdorff space, then is regular so there is a base of closed neighborhoods at each point\ and each of these neighborhoods is compact. Therefore is locally compact.

4) Each ordinal space Ò!ßαα Ñ is locally compact. The space Ò!ß Ó is a (one-point) compactification of Ò!ßαα Ñ iff is a limit ordinal.

5Ñ\ Example 1.5 shows that if a space has a one-point compactification, it must be locally compact (and, of course, noncompact and Hausdorff). Therefore neither  nor has a one-point compactification. The following theorem characterizes the spaces with one-point compactifications.

Theorem 2.3 A space \\ has a one-point compactification iff is a noncompact, locally compact Hausdorff space. (The one-point compactification of \2 for which the embedding is the identity is denoted \Þ‡ )

Proof Because of Example 1.5, we only need to show that a noncompact, locally compact Hausdorff space \:Â\ has a one-point compactification. Choose a point and let \‡‡ œ \ ∪ Ö:×Þ Put a on \ by letting each point B − \ have its original neighborhood base of compact neighborhoods, and by defining basic neighborhoods of : be the complements of compact subsets of \À

‡‡ U: œÖR©\ À :−R and \ R is compact × .

(Verify that the conditions of the Neighborhood Base Theorem III.5.2 are satisfied.)

‡ If hhU is an open cover of \ and :−Y − , then there exists an R−: with :−R©YÞ ‡ ‡ Since \  R is compact, we can choose Y"8 ß ÞÞÞß Y −h covering \  R . Then ‡‡ ÖY ß Y"8 ß ÞÞÞß Y × is a finite subcover of \ from h . Therefore \ is compact.

\+Á,−\+,‡ is Hausdorff. If , then and can be separated by disjoint open sets in \ and these sets are still open in \O‡ . Furthermore, if is a compact neighborhood of +\O in , then and Ð\‡‡  OÑ are disjoint neighborhoods of + and : in \ .

‡‡ Finally, notice that Ö:× is not open in \  or else Ö:× − F: and then \  Ö:× œ \ would be compact. Therefore every open set containing :\\\ intersects , so is dense in ‡ .

420 Therefore \\ñ‡ is a one-point compactification of .

What happens if the construction for \‡ in the preceding proof is carried out starting with a space \\ which is already compact? What happens if is not locally compact? What happens if \ is not Hausdorff ?

Corollary 2.4 A locally compact Hausdorff space \ is Tychonoff.

Proof \\ is either compact or has a one-point compactification \Þ\‡ Either way, is a subspace of a compact X\# space which (by Theorem VII.5.9) is Tychonoff. Therefore is Tychonoff. ñ

The following theorem about locally compact spaces is often useful.

Theorem 2.5 Suppose E©\ , where \ is Hausdorff.

a) If \EœJ∩KJE\E is locally compact and where is closed and is open in , then is locally compact. In particular, an open (or, a closed) of a locally \ is locally compact.

b) If E\EE is a locally compact and is Hausdorff, then is open in cl\ .

c) If E\EœJ∩KJ is a locally compact subspace of a Hausdorff space , then where is closed and E\ is open in .

Proof a) It is easy to check that if JK is closed and is open in a \ , then JKand are locally compact. It then follows easily that J∩K is also locally compact. ( Note: Part a) does not require that \ be Hausdorff.)

b) Let +−E and let O be a compact neighborhood of + in E . Then +− intE OœY . Since E is Hausdorff, O is closed and therefore Y© clEE Y©O , so cl Y is compact.

Because YEßZ\E∩ZœY is open in there is an open set in with and we have:

cl(\\E∩ZÑ∩EœÐ clcl YÑ∩EœE Y ©E so ÐE∩ZÑ∩E cl\ ( ) is compact and therefore closed in \ (since \ is Hausdorff).

SinceE∩Z ©Ð cl\\\ ( E∩ZÑ ) ∩Eß we have cl ÐE∩ZÑ©Ð cl YÑ∩Eœ clE Y ©EÞ

Moreover, since ZZ∩E©ÐZ∩EÑ is open, then cl\\ cl (this is true in any space \ : why? ).

So [ œZ ∩cl\\ E© cl ÐE∩ZÑ©Ð cl \ YÑ∩Eœ clE Y ©EÞ

Then +−[ ©E and [ is open in cl\\ E so +− int cl E . Therefore E is open in cl \ EÞ

421 c) Since EEEEœE∩K is locally compact, part b) gives that is open in cl\\ , so cl for some open set K\ in . Let JœEÞñ cl\

Corollary 2.6 A dense locally compact subspace of a Hausdorff space \\Þ is open in

Proof This follows immediately from part b) of the theorem ñ

Corollary 2.7 If \\ is a locally compact, noncompact Hausdorff space, then is open in any compactification ]\ that contains .

Proof This follows immediately from Corollary 2.6. ñ

Corollary 2.8 A locally compact Ð\ß .Ñ is completely metrizable. µµµ Proof Let Ð\ ß . Ñ be the completion of Ð\ß .ÑÞ \ is locally compact and dense in \ so µµ \\ is open in . Therefore \K\ is a $ -set in so it follows from Theorem IV.7.5 that \ is completely metrizable. ñ

Theorem 2.9 Suppose is Hausdorff. Then is locally compact iff \œα−E \α Ág \

i) each \α is locally compact ii) \−EÞα is compact for all but at most finitely many α

Proof Assume \Y\B−Y is locally compact. Suppose αααα is open in and . Pick a point " D−1α ÒYÓαααwith D œB. Then D has a compact neighborhood O in \ for which " D−O©11α ÒYÓαα. Since is an open continuous map, 1 α ÒOÓ is a compact neighborhood of B α with B−αα1 ÒOÓ©YÞ α Therefore \ α is locally compact, so i) is true.

To prove ii), pick a point B−\ and let O be a compact neighborhood of B . Then

B − Y œ  Yαα"8 ß ÞÞÞß Y  © O for some basic open set Y . If αα Á"8 ,..., α ß we have 11ααÒOÓ ª ÒY Ó œ \ α. Therefore \ α is compact if ααα Á"8 ,..., Þ

Conversely, assume i) and ii) hold. If B−Y ©\ , where Y is open, then we can choose a basic open set Z œ  Zαα"8 ß ÞÞÞß Z  so that B − Z © Y . Without loss of generality, we can assume that \Áα is compact for αα"8 ,..., α (why? ). For each 3 we can choose a compact neighborhood Oαα33 of B so that B α 3 − O αα 33 © Z © \ α 3Þ Then  O α " ß ÞÞÞß O α 8  œ O ‚ ÞÞÞ ‚ O ‚ \ \ is a compact neighborhood of B and αα"8 ααÁ"8 ßÞÞÞ α α B −  Oαα"8 ß ÞÞÞß O  ©  Z αα "8 ß ÞÞÞß Z  © Y. So \ is locally compact. ñ

422 3. The Size of Compactifications

Suppose \\©]]\ is a Tychonoff space, that , and that is a compactification of . How large can l]  \l be? In all the specific example so far, we have had l]  \l œ " or l]  \l œ #Þ

Example 3.1 This example illustrates a compactification of a discrete space created by adding - points.

Let M!" œ ÖÐBß !Ñ À B − Ò!ß "Ó× and M œ ÖÐBß "Ñ À B − Ò!ß "Ó× , two disjoint “copies” of Ò!ß "ÓÞ Let Define a topology on ]œM∪M!" by using the following neighborhood bases:

i) points in M"" are isolated: for : − M ß a neighborhood base at : is U : œ ÖÖ:××

ii) if :œÐBß!Ñ−M! : a basic neighborhood of : is any set of form Z ∪ ÖÐDß "Ñ À ÐDß !Ñ − Z ß D Á B× , where Z is an open neighborhood of : in Ò!ß "Ó

(Check that the conditions in the Neighborhood Base Theorem III.5.2 are satisfied.)

] is called the “double” of the space Ò!ß "Ó œ M! Þ

Clearly, ]] is Hausdorff, and we claim that is compact. It is sufficient to check that any covering h of ] by basic open neighborhoods has a finite subcover.

Let jhœÖ[− À [∩M!! Ág×Þ j covers M and each [ − j has form Z ∪ ÖÐDß "Ñ À ÐDß !Ñ − Z ß D Á B×, where Z is open in M! . Clearly, the open “ Z -parts” of the sets in jj cover the compact space M!"8! , so we finitely many [ ß ÞÞÞß [ − cover M . These sets also cover M""5"3 , except for possibly finitely many points : ß ÞÞÞß : − M . For each such point : choose a set Y33"8"5 −hh containing : . Then Ö[ ß ÞÞÞß [ ß Y ß ÞÞÞß Y × is a finite subcover from ..

Every neighborhood of a point in MMMœ]]!"" intersects , so cl . Therefore is a compactification of the discrete space Ml]Mlœ-Þ"" and

‡ Since MM"" is locally compact, also has another quite different compactification M" for which ‡ lM"  M" l œ "Þ In fact, it is true (depending on \Ñ that can be many different compactifications ]l]\l, each with a different size for .

But, for a given space \] and a compactification , there is an upper bound for how large l]  \l can be. We can find it using the following two lemmas.

Recall that the weight AÐ] Ñ of a space Ð] ßgUU Ñ is defined by AÐ] Ñ œ i!  min Öl l À is a base for g ×Þ (Example VI.4.6 )

AÐ] Ñ Lemma 3.2 If ]X is a ! space, then l]lŸ#Þ

Proof Let UUU be any base for ]C−]ßœÖY−ÀC−Y×Þ] , and for each point let C Since w is XÁCÁCCÄ©!CC , we have UUw if . Therefore the map UU C is one-to-one, so

423 l] l Ÿ lcU Ð Ñl œ #llU . In particular, if we pick U to be a base with the least possible , minimal cardinality, then l] l Ÿ #||U Ÿ 2AÐ] Ñ Þ ñ

Lemma 3.3 Suppose ]X is an infinite $ space and that \ is a dense subspace of ] . Then AÐ]ÑŸ#l\l Ÿ# l] l Þ

Proof A X]$ space with a finite base must be finite, so every base for must be infinite. Let UαœÖYααααα À −E× be a base for ] . Each Y is open so we have i) Y © int cl Y © cl Y , and ii) because \]YœÐY∩\ÑÐÞ is dense in , clαα cl see Lemma IV.6.4Ñ

For each α , define ZœÐY∩\ÑY©YœÐY∩\ÑœZÞαα int cl , so that αααα int cl int cl

w Then UαœÖZα À −E× is also a base for ] : to see this, suppose C−S©] where S is open. By regularity, there is a Yααααα such that C−Y © int cl Y œZ © cl Y ©S .

Since each Y∩\©\αα , there are no more distinct Z 's than there are subsets of \ , that is lUcww l Ÿ l Ð\Ñl. Since U must be infinite, we have AÐ] Ñ Ÿ l Uc wl\ll]l l Ÿ l Ð\Ñl œ # Ÿ # Þ ñ

Theorem 3.4 If ]\H\l]lŸ#Þ is a compactification of and is dense in , then #lHl

Proof ] is Tychonoff. If ] is finite, then Hœ\œ] so l]lŸ###l] l œ# lHl . Therefore we can assume ]H]AÐ]ÑŸ# is infinite. Since is dense in , lHl (by Lemma 3.3), and therefore so l] l Ÿ #AÐ] Ñ Ÿ # #lHl (by Lemma 3.2) ñ

Example 3.5 An upper bound on the size of a compactification of  is #œ##-i! . More generally, a compactification of any separable Tychonoff spaceß such as  ,  or ‘ can have no more than #- points.

We will see in Section 6 that Theorem 3.4 is “best possible” upper bound. For example, there actually exists a compactification of " , called , with cardinality #œ#x#-i! (It is difficult to imagine how the “tiny” discrete set " can be dense in a such a large compactification .

Assume such a compactification " exists. Since  is dense, each point 5 in "  is the limit of a net in 5 , and this net has a universal subnet which converges to .

Since " is Hausdorff, a universal net in  has at most one limit in "  , so there are at least as many universal nets in " as there are points in # , namely - . None of these universal nets can be trivial (that is, eventually constant). Therefore each of these universal nets is associated with a free ( œ nontrivial) ultrafilter in  . So there must be #- free ultrafilters in  .

424 4. Comparing Compactifications

We want to compare compactifications of a Tychonoff space \ . We begin by defining an equivalence relation ¶\ between compactifications of . Then we define a relation . It will turn out that \Þ can also be used to compare equivalence classes of compactifications of When applied in a set equivalence classes of compactifications of \ , will turn out to be a partial ordering.

The definition of ¶ requires that we use the formal definition of a compactification as a pair.

Definition 4.1 Two compactifications Ð]"" ß 2 Ñ and Ð] ## ß 2 Ñ of \ are called equivalent , written Ð]"" ß 2 Ñ ¶ Ð] ## ß 2 Ñ, if there is a homeomorphism 0 of ] " onto ] # such that 0 ‰ 2 " œ 2 # Þ

In the special case where \©]\©]"#"# , , and 2œ2œ the identity map on \ , then the condition 0œ0‰2"# œ2 simply states that 0ÐBÑœB for B−\ that is, points in \ are fixed under the homeomorphism 0 .

It is obvious that Ð]"" ß 2 Ñ ¶ Ð] "" ß 2 Ñ and that ¶ is a transitive relation among compactifications " of \ . Also, if Ð]ß2ѶÐ]ß2Ñ"" ## , then 0 À] # Ä] " is a homeomorphism and " " 0 ‰ 2#""##"" œ 0 Ð0 ‰ 2 Ñ œ 2 so that Ð] ß 2 Ñ ¶ Ð] ß 2 Ñ . Therefore ¶ is a symmetric relation, so¶\ is an equivalence relation on any set of compactifications of .

Example 4.2 Suppose \ is a locally compact, noncompact Hausdorff space. We claim that all one-point compactifications of \¶ are equivalent. Because is transitive, it is sufficient to show that each one-point compactification Ð]"" ß 2 Ñ is equivalent to the one-point compactification Ð]‡ ß 3Ñ constructed in Theorem 2.3.

‡‡ Let ] œ \ ∪ Ö:× and ]""  2 Ò\Ó œ Ö: " ×Þ Define 0 À ] Ä ] " by

2ÐCÑif C−\ 0ÐCÑ œ "  :Cœ:" if

425 00‰3œ2Þ0 is clearly a bijection and " We claim is continuous.

w If C−\À Let Z be an open set in ]"" with 0ÐCÑœ2ÐCÑ−ZÞ Then Z œZ Ö:× " " is also open in 2""" Ò\ÓÞ Since 2 À \ Ä 2 Ò\Ó is a homeomorphism, Y œ 2" ÒZ " Ó is open ‡‡w in \\ and is open in ] . Then C−YY , is open in ] and 0ÒYÓœ2ÒYÓœZ©ZÞ" Therefore 0C is continuous at .

If Cœ:À Let Z be an open set in ]"""" with 0Ð:Ñœ: −ZÞ Then ] Z œO is a " ‡ compact in 2Ò\Ó"" , so 2" ÒOÓœO is a compact (therefore closed) set in ] Þ Then Yœ]‡ O is a neighborhood of : and 0ÒYÓ©ZÞ Therefore 0 is continuous at : .

Since 0X00 is a continuous bijection from a compact space to a # space, is closed and therefore is a homeomorphism.

Therefore (up to equivalence) we can talk about the one-point compactification of a noncompact, locally compact Hausdorff space \ . Topologically, it makes no difference whether we think of the one-point compactification of ‘ geometrically as W:" , with the North Pole as the “,” or whether we think of it more abstractly as the result of the construction in Theorem 2.3.

Question: Are all two point compactifications of Ð  "ß "Ñ equivalent to Ò  "ß "Ó ?

Example 4.3 Suppose Ð]"" ß 2 Ñ is a compactification of \Þ Then Ð] "" ß 2 Ñ is equivalent to a compactification Ð] ß 3Ñ where \ © ] and 3 is the identity map. We simply define ] œ Ð]""  2 Ò\ÓÑ ∪ \, topologized in the obvious way  in effect, we are simply giving each point 2ÐBÑ"" in ] a new “name” B . We can then define 0À] " Ä] by

DD−]2Ò\Óif "" 0ÐDÑ œ " "  3‰2"" ÐDÑœ2 ÐDÑif D−2" Ò\Ó

Clearly, 0‰2""" œ3, so Ð]ß2ѶÐ]ß3Ñ.

Example 4.3 shows means that whenever we work with only one compactification of \ , or are discussing properties that are shared by all equivalent compactifications of \ , we might as well (for simplicity) replace Ð]"" ß 2 Ñ with an equivalent compactification ] where ] contains \ as a dense subspace.

Example 4.4 Homeomorphic compactifications are not necessarily equivalent. In this example we see two dense embeddings 2ß2"# of  into the same compact Hausdorff space ] that produce nonequivalent compactifications.

" # Let ] œ ÖÐ8 ß 3Ñ À 3 œ "ß # and 8 −‘ × ∪ ÖÐ!ß "Ñß Ð!ß #Ñ× © .

" 2" Ð#8Ñ œ Ð8 ß "Ñ Let 2À"" Ä] by" . Ð]ß2Ñ is a 2-point compactification of .  2" Ð#8  "Ñ œ Ð8 ß #Ñ

426 " th 2# Ð8Ñ œ Ð4 ß "Ñif 8 is the 4 element of Ö"ß #ß %ß &ß (ß )ß "!ß ""ß ÞÞÞ× Let 2À#  Ä] by " th  2# Ð8Ñ œ Ð4 ß #Ñif 8 is the 4 element of Ö$ß 'ß *ß "#ß ÞÞÞ×

"" For example, 2## Ð(Ñ œ Ð&$ ß "Ñ and 2 Ð*Ñ œ Ð ß #ÑÞ Ð] ß 2 # Ñ is also a two-point compactification of .

Topologically, each compactification is the same space ]Ð]ß2ÑÐ]ß2Ñ , but "# and are not equivalent compactifications of  :

Suppose 0À]Ä] is any (onto) homeomorphism.

Ð2"" Ð#8ÑÑ Ä Ð!ß "Ñ , so 0Ð2 Ð#8ÑÑ Ä 0ÐÐ!ß "ÑÑ , and 0ÐÐ!ß "ÑÑ œeither Ð!ß "Ñ or Ð!ß #Ñ Ðwhy? Ñ.

"""""" But the sequence Ð2# Ð#8ÑÑ œ ÐÐ#$#'(% ß "Ñß Ð ß "Ñß Ð ß #Ñß Ð ß "Ñß Ð ß "Ñß Ð ß #Ñß ÞÞÞ Ñ . does not converge to either Ð!ß "Ñ or Ð!ß #Ñ . Therefore 0 ‰ 2"# Á 2 , so Ð]"" ß 2 Ñ ¶ÎÐ]ß2Ñ ##

By adjusting the definitions of h"# and 2 , we can create infinitely many nonequivalent 2-point compactifications of  all using different embeddings of into the same space ] .

427 We now define a relation \ between compactifications of a space .

Definition 4.5 Suppose Ð]## ß 2 Ñ and Ð] "" ß 2 Ñ are compactifications of \ . We say that Ð]## ß 2 Ñ Ð] "" ß 2 Ñ if there exists a continuous 0 À ]2 Ä ] " such that 0 ‰ 2 # œ 2 " Þ

Notice that:

i) Such a mapping 0]0Ò]Ó] is necessarily onto "# : is compact and therefore closed in " ; so 0Ò]### Ó œcl 0Ò] Ó ª cl 0Ò2 Ò\ÓÓ œ cl 2 "" Ò\Ó œ ] Þ

ii) If \©]ß\©]# " and 2 "# œ2 œ the identity map 3 , then the condition 0‰2 #" œ2 simply states that 0l\ œ 3 .

iii) 0Ò]2Ò\ÓÓ©]2Ò\ÓÀ## "" that is, the “points added” to create ] # are mapped onto the “points added” to create ]"## . To see this, let D − ]  2 Ò\ÓÞ We want to show 0ÐDÑ − ]""  2 Ò\ÓÞ So suppose that 0ÐDÑ œ 2 " ÐBÑ − 2 " Ò\ÓÞ

Since 2Ò\Ó## is dense in ] , there is a net in 2Ò\Ó # converging to DÀ

Ð2# ÐB- ÑÑ Ä D ЇÑ

0 is continuous, so 2ÐBÑœ0Ð2ÐBÑÑÄ0ÐDÑœ2ÐBÑ"#-- " . But 2À\Ä2Ò\Ó"" is a homeomorphism so

" " ÐB-- Ñ œ Ð2"" 2"" ÐB ÑÑ Ä 2 2 ÐBÑ œ B − \ß and therefore Ð2### ÐB- ÑÑ Ä 2 ÐBÑ − 2 Ò\Ó Ð‡‡Ñ

A net in ]# has at most one limit, so Ð‡Ñ and Ї‡Ñ give that D œ 2# ÐBÑ . This is impossible since DÂ2Ò\ÓÞ#

iv) From iii) we conclude that if Ð]## ß 2 Ñ Ð] "" ß 2 Ñ , then l] #  2 # Ò\Ól l] "  2 " Ò\Ól

428 Suppose Ð]## ß 2 Ñ Ð] "" ß 2 Ñ . The next theorem tells us that the relation “ ” is unaffected if we replace these compactifications of \ with equivalent compactifications so we can actually compare equivalence classes of compactifications of \ by comparing representatives of the equivalence classes. The proof is very easy and is omitted.

Theorem 4.6 Suppose Ð]## ß 2 Ñ and Ð] "" ß 2 Ñ are compactifications of \ and that ww ww ww ww Ð]## ß 2 Ñ Ð] "" ß 2 Ñ. If Ð] ## ß 2 Ñ ¶ Ð]## ß 2 Ñ and Ð] "" ß 2 Ñ ¶ Ð] "" ß 2 Ñ , then Ð] ## ß 2 Ñ Ð] "" ß 2 ÑÞ

The ordering “ \Þ ” is well behaved on the equivalence classes of compactifications of

Theorem 4.7 Let VV be a set of equivalence classes of compactifications of \Ðß Ñ . Then is a poset.

Proof It is clear from the definition that is both reflexive and transitive. We need to show that is also antisymmetric. Suppose ÒÐ]"# ß 3ÑÓ and ÒÐ] ß 3ÑÓ are equivalence classes of compactifications of \ (By Theorem 4.6, we are free to choose from each equivalence class representative compactifications with \©]+8.3 embeddings 2œ3œ the identity map ).

If both Ð]"# ß 3Ñ Ð] ß 3Ñ and Ð] #" ß 3Ñ Ð] ß 3Ñ hold, then we have the following maps:

with 0 ‰ 3 œ 3 œ 1 ‰ 3 . For B − \ , 1Ð0ÐBÑÑ œ 1Ð0Ð3ÐBÑÑ œ 1Ð3ÐBÑÑ œ 3ÐBÑ œ B , so the maps 1‰0 and the identity 3À]"" Ä] agree on the dense subspace \ . Since ] " is Hausdorff, it follows that 1‰0 œ3 everywhere in ]" Þ (See Theorem 5.12 in Chapter II, and its generalization in Exercise E9 of Chapter III.) Similarly 0‰1 and 3À]## Ä] agree on the dense subspace \ so 0‰1œ3 on ]Þ#

429 Since 0‰1œ3 and 1‰0œ3ß0 and 1 are inverse functions and 0 is a homeomorphism. Therefore Ð]"# ß 3Ñ ¶ Ð] ß 3Ñ . So we have shown that if ÒÐ] "# ß 3ÑÓ ÒÐ] ß 3ÑÓ and ÒÐ] "# ß 3ÑÓ Ÿ ÒÐ] ß 3Ó , then ÒÐ]"# ß 3ÑÓ œ ÒÐ] ß 3Ó ñ

An equivalence class of compactifications of a space \ is “too big” to be a set in ZFC . ÐIt is customary to refer informally to such collections “too big” to be sets in ZFC as “classes.”Ñ

However, suppose Ð] ß 3Ñ represents one of these equivalence classes. If \ has weight 7 , then \ contains a H with lHlŸ7 . It follows from Lemma 3.3 that AÐ]ÑŸ#7 so, by Theorem VII.3.17, ] can be embedded in the Ò!ß "Ó#7 . Therefore every compactification of \ can be represented by a subspace of the one fixed cube Ò!ß "Ó#7 .

Therefore we can form a set consisting of one representative from each equivalence class of compactifications of \À this set is just a certain set of subspaces of Ò!ß"Ó#7 . This set is partially ordered by .

In fact, we can even given a bound on the number of different equivalence classes of compactifications of \\ : since every compactification of can be represented as a subspace of Ò!ß "Ó#7 , the number of equivalence classes of compactifications of \ is no more than 77 7 |cÐÒ!ß "Ó#Ð-Ñ#7## Ñl œ # œ # . In other words, there are no more than ### different compactifications of \ .

Example 4.8 Let Ð]"" ß 2 Ñ be a 1-point compactification of \ . For every compactification Ð] ß 2Ñ of \Ð]ß2Ñ Ð]ß2Ñ , "" . (So, among equivalence classes of compactifications of \ , the equivalence class ÒÐ]"" ß 2 ÑÓ is smallest.)

By Theorem 4.6, we may assume \©]ß\©]"" and that 22 , are the identity maps; in fact, ‡ we may as well assume ]œ\Ð" the one-point compactification constructed in Theorem 2.3 Ñ .

Since \\ÐÑÞ has a one-point compactification, is locally compact see Example 1.5 By Corollary 2.7, \]\Þ is open in both and ‡

Let \\œÖ:ׇ and define

CC−\if 0À]Ä\‡ by 0ÐCÑœ  :D−]\if

To show that Ð] ß 3Ñ Ð\‡ ß 3Ñ , we only need to check that 0 is continuous each point D − ] .

If C−\ and Z is an open set containing 0ÐCÑœC in \‡ , then C−Y œZ Ö:× which is open in \ and therefore also open in ] . Clearly, 0ÒYÓœY ©ZÞ

If D−]\ and Z is an open neighborhood of 0ÐDÑœ: in \߇‡ then \ Z œO is a compact subset of \Þ Therefore O is closed in ] so Y œ ]  O is an open set in ]D0ÒYÓ©ZÞ containing and

430 5. The Stone-Cech Compactification

Example 4.8 shows that the one-point compactification of a space \ , when it exists , is the smallest compactification of \\ . Perhaps it is surprising that every Tychonoff space has a largest compactification and, by Theorem 4.7, this compactification is unique up to equivalence. In other words, a poset which consists of one representative of each equivalence class of compactifications of \ has a largest (not merely maximal!) element. This largest compactification of \ is called the Stone-Cech (pronounced “check”) compactification and is denoted by "\ .

Theorem 5.1 1) Every Tychonoff Space \ has a largest compactification, and this compactification is unique up to equivalence. (We may represent the largest compactification by Ð\ß3Ñ"" where \ª\ and 3 is the identity map. We do this in the remaining parts of theorem.)

2) Suppose \] is Tychonoff and that is a compact Hausdorff space. Every continuous 0À\Ä] has a unique continuous extension 0" À" \Ä] . (The extension 0 " is called the Stone extension of 0\ . The property of " in 2) is called the Stone Extension Property. )

3) Up to equivalence, "\\ is the only compactification of with the Stone Extension Property. (In other words, the Stone Extension Property characterizes "\ among all compactifications of \ .)

Example 5.2 (assuming Theorem 5.1) Ò!ß "Ó is a compactification of Ð!ß "Ó . However the " 0 À Ð!ß "Ó Ä ] œ Ò  "ß "Ó given by 0ÐBÑ œ sin ÐB Ñ cannot be continuously extended to a map 0" À Ò!ß "Ó Ä ] . Therefore Ò!ß "Ó Á""‘ Ð!ß "Ó . Is it possible that W" œ ?

Proof of Theorem 5.1

1) Since \\ is Tychonoff, has at least one compactification. Let ÖÐ]ß3ÑÀ−E×αα α be a set of compactifications of \\©]ß3À\Ä] , where αα α is the identity, and ] α is chosen from each equivalence class of compactifications of \ . (As noted in the remarks following Theorem 4.7, this is a legitimate set since every compactification of \ can be represented as subset of one fixed cube Ò!ß "Ó5 .)

Define by . This “diagonal” map sends /À\Ä] œ Ö]α Àαα −E× /ÐBÑÐÑœ3ÐBÑœB / each BB/ to the point in the product all of whose coordinates are , and is the evaluation map generated by the collection of maps 3À\Ä]Þ\αα is a subspace of ]ß α and the is precisely the induced on \3 by each α (see Example VI.2.5 ). It follows from Theorem VI.4.4 that /\ is an embedding of into the compact space ] . If we define "\œ cl] /Ò\Ó , then Ð" \ß /Ñ is a compactification of \ .

Every compactification of \Ð]ß3Ñ\ is equivalent to one of the αα . Therefore, to show " is the largest compactification we need only show that Ð"α \ß/Ñ Ð]αα ß3 Ñ for each −E . This, however, is clear: in the diagram below, simply let 0œαα1" l\Þ

431

Then 0ααα ‰ / œ 3 because 0 Ð/ÐBÑÑ œ1α Ð/ÐBÑÑ œ /ÐBÑÐ Ñ œ 3ÐBÑ œ BÞ

Therefore Ð" \ß/Ñ Ð]α ß3Ñ.

Note: now that the construction is complete, we can replace Ð\ß/Ñ" with an equivalent largest compactification actually containing \À Ð" \ß3ÑÞ

Since Ð]ß3Ñ is antisymmetric among the compactifications αα , the largest compactification of \ is unique (up to equivalence).

2) Suppose 0À\Ä] where ] is a compact Hausdorff space. First, we need to produce a continuous extension 0À\Ä]Þ" "

Define 1 À \ Ä" \ ‚ ] by 1ÐBÑ œ ÐBß 0ÐBÑÑÞ Clearly, 1 is "  " and continuous ß and \ has the weak topology generated by the maps 3À\Ä\ and 0À\Ä] , so 1 is an embedding. Since "\‚] is compact, Ð cl"\‚] 1Ò\Óß1Ñ is a compactification of \ .

ButÐ"" \ß 3Ñ Ð cl 1Ò\Óß 1Ñ , so we have a continuous map 2 À \ Ä cl 1Ò\Ó for which 2 ‰ 3 œ 1 2ÐBÑœ1ÐBÑB−\ that is for (see the following diagram )

432

"" For D −"1 \ , define 0 ÐDÑ œ] ‰ 2ÐDÑÞ Then 0 is continuous and for B − \ we have " " 0 ÐBÑ œ11]] Ð2ÐBÑÑ œ Ð2Ð3ÐBÑÑÑ œ 11 ]] Ð1ÐBÑÑ œ ÐBß 0ÐBÑÑ œ 0ÐBÑ, so 0 l\ œ 0 .

If 5À" \Ä] is continuous and 5l\œ0 , then 5 and 0"" agree on the dense set \ , so 5œ0 Therefore the Stone extension 0Ð" is unique. See Theorem II.5.12, and its generalization in exercise E9 of Chapter III..)

3) Suppose Ð] ß 3Ñ is a compactification of \ with the Stone Extension Property. Then the identity map 3À\Ä"" \ has an extension 3‡‡ À] Ä \ such that 3 ‰3œ3 , so Ð] ß 3Ñ Ð"" \ß 3Ñ. Since \ is the largest compactification of \ , Ð] ß 3Ñ ¶ Ð " \ß 3Ñ . ñ

The Tychonoff Product Theorem is equivalent to the Axiom of Choice AC (see Theorem IX.6.5 ). Our construction of "\ used the Tychonoff Product Theorem but only applied to a collection of compact Hausdorff spacesÞ In fact, as we show below, the existence of a largest compactification "\X is equivalent to “the Tychonoff Product Theorem for compact # spaces.”

The “Tychonoff Product Theorem for compact X# spaces” also cannot be proven in ZF, but it is strictly weaker than AC. (In fact, the “Tychonoff Product Theorem for compact X# spaces” is equivalent to a statement called the “Boolean Prime Ideal Theorem.”)

The main point is that the very existence of "\ involves set-theoretic issues and any method for constructing "\ must, in some form, use something beyond ZF set theory something quite close to the Axiom of Choice.

Theorem 5.3 If every Tychonoff space \\ has of a largest compactification " , then any product of compact Hausdorff spaces is compact.

433 Proof Suppose is a collection of compact spaces. Since is Ö\α Àα − E× X# α−E \α Tychonoff, it has a compactification and for each the projection map can be "α1Ò\Óα−E αα extended to " 1"α ÀÒα−E \ÓÄ\Þαα

For each , define a point with coordinates " . B −"α1 Òα−E \α Ó 0ÐBÑ − \ 0ÐBÑÐ Ñ œα ÐBÑ

0À" Òαα−E \ÓÄαα −E \ is continuous because each coordinate function " is continuous. If 0B−\1α α−E α , then " for each , so ©"α11αα Òα−E \ααα Ó 0ÐBÑÐ Ñ œα ÐBÑ œ ÐBÑ œ BÐ Ñ œ B Therefore is a continuous of the compact space , so 0ÐBÑ œ BÞα−E \α " Òα−E \α Ó is compact. α−E\ñα

We want to consider some other ways to recognize ""\\ . Since can be characterized by the Stone Extension Property, the following technical theorem about extending continuous functions will be useful.

Theorem 5.4 (Taimonov) Suppose C is a dense subspace of a Tychonoff space \] and let be a compact Hausdorff space. A continuous function 0ÀGÄ] has a continuous extension µ 0À\Ä] iff

" " whenever EF and are disjoint closed sets in ] , cl\\ 0ÒEÓ∩0ÒFÓœgÞ cl

µµµ" " Proof Ê0: If exists and EF and are disjoint closed sets in ] , then 0ÒEÓ∩0ÒFÓœg . " " µµ" " But these sets are closed in \ , so 0 ÒEÓ œ cl\\ 0 ÒEÓ and 0 ÓFÓ œ cl 0 ÒFÓ . Therefore " " cl\\0ÒEÓ∩0ÒFÓœg cl . µµ µ É0À\0Gœ00 : We must define a function Ä] such that | and then show that is

434 continuous. For B−\ , let aB be its neighborhood in \ . Define a collection of closed sets YB in ] by YaBBœÖ cl 0ÒG∩YÓÀU − ×

Then cl0ÒG ∩Y"# Ó∩ cl 0ÒG ∩Y Ó ª cl0ÒG ∩Y"# ∩Y ] Á g (since G is dense in \ ). Therefore is a family of closed sets in with the finite intersection property so (because is YYB ]Ág] B compact) .

We claim that contains only one point: for some . YYBBœÖC× C−]

Suppose Cß D −YB$ . If C Á D , then (since ] is X ) we can pick open sets Y, Z so that  " " C−Y and D−Z and cl Y∩ cl Z œg . Then cl\\ 0 Ò cl YÓ∩ cl 0 Ò cl ZÓœg so, of " " course, cl\\0ÒYÓ∩0ÒZÓœg cl . Taking complements, we get

" " Ð\  cl\\ 0 ÒY ÓÑ ∪ Ð\  cl 0 ÒZ ÓÑ œ \

" so B is in one of these open sets: say B − [ œ \  cl\B 0 ÒY ÓÞ Since [ − a , cl0ÒG ∩[Ó −YB . We claim cl 0ÒG ∩[Ó © ] Y , from which will follow the contradiction that . CÂYB

" To check this inclusion, simply note that G∩[œG cl\ 0 ÒYÓ . Therefore, " " if?−G∩[ , we have ?  cl\ 0 ÒY Ó , so ?  0 ÒY Ó , so 0Ð?Ñ Â Y . Thus, 0Ò CYU ∩[Ó©©  (a closed set) so cl 0Ò CYU ∩[Ó  Þ µµ Define 0ÐBÑœC . We claim that 0 works. µ 0 | Gœ0 : Suppose B−GÞ Ua œÖG∩Y ÀY −B × is the neighborhood filter of B in G so UUaÄB in G. Since 0 is continuous, the filter base 0Ò ÓœÖ0ÒG∩YÓÀY−B × Ä 0 ÐBÑ in ] . In particular, 0ÐBÑ is a cluster point of 0ÒU Ó , so 0ÐBÑ − cl Ð0ÒG ∩ Y ÓÑ µµ  =. So œYB Ö0ÐBÑ× 0ÐBÑœ0ÐBÑÞ µ µ 0B−\Z]Cœ0ÐBÑ−ZÞ is continuous: Let and let be open in with Since , there exist ,..., such that YBBœÖCשZYY"8 −R

cl0ÒG∩Y"8 Ó ∩ ÞÞÞ ∩ cl 0ÒG∩Y Ó©Z

(If Z is an open set in a compact space and Y is a family of closed sets with then some finite subfamily of satisfies . Why? ) YY©Zß J"8 ∩ÞÞÞ∩J ©Z

Let [œY"8∩∩Y... −aB . If D −[ , then µ 0ÐDÑ− cl 0ÒG∩[Ó© cl0ÒG ∩Y"8 Ó∩ ... ∩ cl 0 ÒG ∩Y Ó©Z µµ so 0Ò[Ó©Z . Therefore 0 is continuous at B . ñ

Corollary 5.5 Suppose ]]"# and are compactification of \ where the embeddings are the identity map. Then Ð]"# ß 3Ñ ¶ Ð] ß 3Ñ iff : for every pair of disjoint closed sets in \ ,

435 cl]]""E∩ cl Fœg Í cl ]] ## E∩ cl FœgÑ (*)

Proof If Ð]"# ß 3Ñ ¶ Ð] ß 3Ñ , it is clear that (*) holds. If (*) holds, then Taimonov's Theorem guarantees that the identity maps 3À\Ä]""## and 3À\Ä] can be extended to maps 0À]Ä]"# " and 0À]Ä] #" # . It is clear that 0‰0l\ "# and 0‰0l\ #" are the identity maps on the dense subspace \0‰00‰0 . Therefore "# and #"are each the identity everywhere so 00 "# , are so Ð]"# ß 3Ñ ¶ Ð] ß 3Ñ . ñ

For convenience, we repeat here a definition included in the statement of Theorem VII.5.2

Definition 5.6 Suppose EF and are subspaces of \EF . and are completely separated if there exists 0−GÐ\Ñ such that 0lEœ! and 0lFœ"Þ (It is easy to see that !ß" can be replaced in the definition by any two real numbers +ß , .)

Urysohn's Lemma states that disjoint closed sets in a are completely separated.

Using Taimonov's theorem, we can characterize "\ in several different ways. In particular, condition 4) in the following theorem states that "\ is actually characterized by the “extendability” of continuous functions from \ into Ò!ß "Ó  a statement which looks weaker than the full Stone Extension Property.

Theorem 5.7 Suppose Ð] ß 3Ñ is a compactification of \ , where ] ª \ and 3 is the identity. The following are equivalent:

1) ]\ is " (that is, ] is the largest compactification of \ ) 2) every continuous 0À\ÄO , where O is a compact Hausdorff space, can be µ extended to a continuous map 0 À] ÄO µ 3) every continuous 0À\ÄÒ+ß,Ó can be extended to a continuous 0 À] ÄÒ+ß,Ó µ 4) every continuous 0À\ÄÒ!ß"Ó can be extended to a continuous 0 À ] Ä Ò!ß "Ó 5) completely in \] have disjoint closures in 6) disjoint zero sets in \] have disjoint closures in 7) if ^^"# and are zero sets in \ , then cl ]"#]"]# Ð^∩^Ñœ^∩^ß cl cl

Proof Theorem 5.1 gives that 1) and 2) are equivalent, and the implications 2)ÊÊ 3) 4) are trivial.

4)ÊEF 5) If and are completely separated in \ , then there is a continuous 0 À \ Ä Ò!ß "Ó with 0lE œ ! and 0lF œ " . By 4), 0 extends to a continuous map µµµµµ" " " " 0À ] Ä Ò!ß "ÓÞ Then cl]] E©0 cl Ð!Ñ œ 0 Ð!Ñ and cl ]] F©0 cl Ð"Ñ œ 0 Ð"Ñ , so cl]]E∩ cl FœgÞ

5)Ê 6) Disjoint zero sets ^Ð0Ñ and ^Ð1Ñ in \ are completely separated (for example, 0 # by the function 2œ01## ) and therefore, by 5), have disjoint closures in ] .

6Ê\^B−^Þ 7) A zero set neighborhood of is a zero set with int It is easy to show that in a Tychonoff space \BB , the zero set neighborhoods of form a neighborhood base at ().check this!

436 Suppose ^^"# and are zero sets in \ . Certainly, cl ]"#]"]# Ð^∩^Ñ©^∩^ cl cl , so suppose B−cl]" ^ ∩ cl ]# ^Þ If Z is a zero set neighborhood of B , then B− cl]" Ð^∩ZÑ and B−cl]# Ð^ ∩ZÑ (why? ). ^ " ∩Z and ^ # ∩Z are zero sets in \ and B− cl]" Ð^ ∩ZÑ∩ cl ]# Ð^ ∩ZÑ so, by 6). Ð^ " ∩ZÑ∩Ð^ # ∩ZÑœ^ " ∩^ # ∩Z Ág .

Since every zero set neighborhood ZB of intersects ^∩^"# , and the zero set neighborhoods of B are a neighborhood base, we have B − cl]" Ð^ ∩ ^ # ÑÞ

7)Ê0À\ÄOOXEF 2) Suppose that is continuous. is % so if and are disjoint closed sets in O , there is a continuous 1 À O Ä Ò!ß "Ó such that E © ÖB À 1ÐBÑ œ !× œ ^" and F © ÖB À 1ÐBÑ œ "× œ ^#.

" " " " " " Then 0ÒEÓ©0Ò^Ó"#"# and 0ÒFÓ©0Ò^Ó and 0Ò^Ó and 0Ò^Ó are disjoint zero sets in \. By 7), cl 0" ÒEÓ ∩ cl 0 " ÒFÓ © cl 0 " Ò^ Ó ∩ cl 0 " Ò^ Ó œ cl 0 " Ò^ ∩ ^ Ó ]]"]#]"#] µ œgÞ By Taimonov's Theorem 5.4, 0 has a continuous extension 0 À] ÄOÞ ñ

Example 5.8

1) By Theorem VIII.8.8, every continuous function 0 À Ò!ß= Ñ Ä Ò!ß "Ó is “constant on a µ " tail” so 0 can be continuously extended to 0 ÀÒ!ß=" ÓÄÒ!ß"Ó . By Theorem 5.7, Ò!ß="="" Ó œ Ò!ß Ñ.

In this case the largest compactification of Ò!ß=" Ñ is the same as the smallest compactificationÒ!ßÓ the one-point compactification. Therefore, up to equivalence, =" is the only compactification of Ò!ß=" Ñ .

‡ A similar example of this phenomenon is XœÒ!ßÓ‚Ò!ßÓœX==""! , where ‡ XœX ÖÐ=="! ß Ñ× (see the “Tychonoff plank” in Example VIII.8.10 and Exercise VIII.8.11).

2) The one-point compactification ‡ of is not " because the function !8if is even 0 À Ä Ö!ß "×given by 0Ð8Ñ œ cannot be continuously extended to  "8if is odd µ ‡ " 0 À‘ Ä Ö!ß "×. ( Why? It might help think of (topologically) as Ö8 À 8 − × © ÞÑ

Theorem 5.9 "\\ is metrizable iff is a compact (i.e., iff \ is metrizable and \œ" \ÑÞ

Proof É : Trivial \Ê\X is metrizable is Ê\ : " is metrizable Ê % .  "\ is first countable If \ is not compact, there is a sequence ÐBÑ88 in \ with ÐBÑÄ:−" \\Þ Without loss of generality, we may assume the B8 's are distinct (why? ).

437 Let S œ ÖB"$ ß B ß ÞÞÞß B #8" ß ÞÞÞ× and I œ ÖB #% ß B ß ÞÞÞß B #8 ß ÞÞÞ× . S and I are disjoint closed sets in \ so Urysohn's Lemma gives us a continuous 0 À \ Ä Ò!ß "Ó for which 0lS œ ! and 0lI œ " . """ Let 0À\ÄÒ!ß"Ó" be the Stone Extension of 0 . Then 0Ð:Ñœlim 0ÐBÑ#8" 8Ä∞ "" œlim 0ÐB#8" Ñœ!Á"œ lim 0ÐB #8 Ñœ lim 0 ÐB #8 Ñœ0 Ð:Ñß which is impossible. ñ 8Ä∞ 8Ä∞ 8Ä∞

6. The space "

The Stone-Cech compactification of  is a strange and curious space.

Example 6.1 " is a compact Hausdorff space in which  is a countable dense set. Since " is separable, Theorem 3.4 gives us the upper bound l" lŸ##-i! œ#Þ

On the other hand, suppose 0 À Ä Ò!ß "Ó ∩ is a bijection and consider the Stone extension 0"" À" Ä Ò!ß "ÓÞ Since 0 Ò " Ó is compact, it is a closed set in Ò!ß "Ó and it contains the dense set "" . Therefore 0" Ò ÓœÒ!ß"Ó so we have -Ÿl lŸ#Þ-

A similar argument makes things even clearer. By Pondiczerny's Theorem VI.3.5, there is a countable dense set H©Ò!ß"ÓÒ!ß"Ó . Pick a bijection 0À ÄH and consider the extension 0"" À" Ä Ò!ß "ÓÒ!ß"Ó. Just as before, 0 must be onto. Therefore Ò " Ó lÒ!ß "ÓÒ!ß"Ó l œ - - œ # - .

Combining this with our earlier upper bound, we conclude that llœ#Þ"- " is quite large but it contains the dense discrete set  that is merely countable.

Every set E© is a zero set in so we can write

"œœE∪ÐEÑ cl"  cl " cl "  , and by Theorem 5.7(6) these sets are disjoint. Therefore for each E© ß cl" E is a in " . In particular, each singleton EœÖ8× is open in " (that is, 8 is isolated in " ), so  is open in " . Therefore "  is compact.

At each B−" , there is a neighborhood base UB consisting of clopen neighborhoods:

i) if B −U , we can use B œ ÖÖB×× ii) if B−"   , we can use UB œÖ cl" EÀ E©  and B− cl " E×

If YB is an open set in " containing , we can use regularity to choose an open set [B−[©[©YEœ[∩ such that cl" . If  , then B− cl" Eœ cl " Ð[∩ Ñœ cl " [ ©Y . ()Why?

Definition 6.2 Suppose E©\ . E is said to be G‡‡ -embedded in \ if every 0−G ÐEÑ has a µ continuous extension 0−GÐ\ÑÞ‡

To illustrate the terminology:

438 i) Tietze's Theorem states that every closed subspace of a normal space is G ‡ -embedded.

ii) For a Tychonoff space \\ , " is the compactification (up to equivalence) in which \G is ‡ -embedded.

The following theorem is very useful in working with "\Þ

‡ Theorem 6.3 Suppose E©\©"" \ , and that E is G -embedded in \ . Then cl"\ Eœ E .  Proof If 0 À E Ä Ò!ß "Ó is continuous, then 0 extends continuously to 0 À \ Ä Ò!ß "Ó , and, in  µµ turn, 0 extends continuously to 0 À" \ Ä Ò!ß "Ó . Then 0 l cl"\ E is a continuous extension of 0 to cl""\\EE . Since cl has the extension property in Theorem 5.7 (4), cl " \ EœEñ" .

Example 6.4 Since  is discrete, every E© is G‡ -embedded in and so, by Theorem 6.3, cl"Eœ" E .

Of course if EEœEœEEE is finite, cl" " . But if is infinite, then is homeomorphic to , so cl"Eœ"" E is homeomorphic to .

In particular, if „ and are the sets of even and odd natural numbers, we have „œ∪ß so "œ∪cl" „ cl "  so " is the of two disjoint, clopen copies of itself. It is easy to modify this argument to show that, for any natural number 5 , " can be written as the union of 5 disjoint clopen copies of itself.

If we write ∞ , where each 's are pairwise disjoint infinite subsets of , then we œE5œ" 55 E have cl∞∞ cl , and these sets cl are pairwise disjoint copies of " œEªE"5œ"55 5œ" " " E 5 . Moreover, ∞ cl is dense in since the union contains . (If we choose the 's "5œ" "E5 "  E5 properly chosen, can we have ∞ cl ? Why or why not?) " œ 5œ" "E5

Example 6.5 No sequence Ð85 Ñ in " can converge to a point of  . In particular, the sequence Ð8Ñ has no convergent subsequence in " so " is not sequentially compact.

"Bœ8if Define 0 À Ä Ö!ß "× by 0ÐBÑ œ #5 . Consider the Stone extension  ! otherwise """ 0 À" Ä Ö!ß "×. If Ð8555 Ñ Ä : − "   , then Ð0 Ð8 ÑÑ œ Ð0Ð8 ÑÑ Ä 0 Ð:Ñ − Ö!ß "× , so Ð0Ð85 ÑÑ must be eventually constant  which is false.

Therefore " is an example showing that “compactÊÎ sequentially compact.” (See the remarks before and after corollary VIII.8.5.)

439 Theorem 6.6 Every infinite closed set J in " contains a copy of " and therefore satisfies lJ l œ #- Þ

Proof Pick an infinite discrete set E œ Ö+8 À 8 œ "ß #ß ÞÞÞ× © J . (See Exercise III E9 ). Using regularity, pick pairwise disjoint open sets Z+−Z888 in " with .

Suppose 1 À E Ä Ò!ß "Ó ( 1 is continuous since E is discrete). Define K À Ä Ò!ß "Ó by

1Ð+ Ñfor 5 − ∩ Z KÐ5Ñ œ 88 for ∞  !5−Z 8œ" 8 Extend K to a continuous map K" À" Ä Ò!ß "ÓÞ

The following diagram gives a very “distorted” image of how the sets in the argument are related.

" " We have Kl ∩Zœ1Ð+Ñ88 . Since ∩Z 8 is dense in Z 8 (why? ), we have KlZœ1Ð+Ñ 88 so KlEœ1Þ"

Thus, 1 À E Ä Ò!ß "Ó has an extension K" À" Ä Ò!ß "Ó , so E is G ‡ -embedded in " . By Theorem 6.3, cl"Eœ"" E and since E is a countably infinite discrete space, E is homeomorphic to " .

- Since JEœE©JlJlœ#Þñ is closed, cl" " , so

Theorem 6.6 illustrates a curious property of " ÀÞ there is a “gap” in the sizes of closed subsets That is, every closed set in " is either finite or has cardinality #- no sizes in-between! This “gap in the possible sizes of closed subsets” can sometimes occur, however, even in spaces as nice as metric spaces although not if the Generalized Hypothesis is assumed. (See A.H. Stone, Cardinals of Closed Sets , Mathematika 6 (1959), pp. 99-107.)

440 Example 6.7 " is separable, but its subspace "  is not; "  does not even satisfy the weaker countable chain condition CCC (see Definition VIII.11.4). Specifically, we will show that "-  contains pairwise disjoint clopen (in "   ) subsets, each of which is homeomorphic to "  .

Let ÖR> À > − Ò!ß "Ó× be a collection of - infinite subsets of  with the property that any two have finite intersection. (See Exercise I.E41. ) Let YœÐ>>>>"   Ñ∩ cl" Rœ cl " RRÞ Each YÁg>> (why? ) and Y is a clopen set in "   homeomorphic to "   .

Moreover, the Y> 's are disjoint:

w Suppose >Á> . If D−Y>> ∩Yww , then D− cl" R > ∩ cl " R > . In a X " space, deleting finitely many points from an infinite set EEE does not change the set cl (why? ), so D − cl" ÐR>>>  ÐR ∩ Rwwww ÑÑ and D − cl " ÐR >  ÐR >> ∩ R ÑÑ . But R >>>  ÐR ∩ R Ñ and RÐR∩RÑ>>>ww are disjoint zero sets in  and must have disjoint closures.

An additional tangential observation:

If we choose points B>> − Y and let \ œ ∪ ÖB > À > − Ò!ß "Ó× , then \ is not normal  since a separable normal space cannot have a closed discrete subsetÖB> À > − Ò!ß "Ó× of cardinality - . (See the “counting continuous functions” argument in Example VII.5.6.)

The following example shows us that countable compactness and pseudocompactness are not even finitely productive.

Example 6.8 There is a \\‚\ for which is not pseudocompact (so \‚\ is also not countably compact).

Let „œ Ö#ß %ß 'ß ÞÞÞ× and  œ Ö"ß $ß &ß ÞÞÞ× and write " œ cl" „ ∪ cl "  œ "„ ∪ " Þ "„ and " are disjoint clopen copies of " . Choose any homeomorphism 0À "„Ä " 0ÐBÑif B − "„ (necessarily, 0Ò„ Ó œ : why? ) and define 1À" Ä " by 1ÐBÑœ Þ  0ÐBÑB−" if "

The map 111 is a homeomorphism since and " are continuous on the two disjoint clopen sets "„ and " whose union is " . Clearly, 1l  À  Ä  , 1 has no fixed points, and 1 ‰ 1 is the identity map.

Let V"œÖE© ÀE is countably infinite × . l V lœÐ#Ñ-i! œ# - . Let - be the first ordinal with - cardinality #ÖEÀ×ÞlEl and index Vα-α as α"α For each , cl is an infinite closed set so, by - Theorem 6.6, lElœ# cl" α . Therefore cl " EEÁgÞ α α

Pick :EE!!! to be a of not in . Proceeding inductively, assume that for all α"- we have chosen a limit point :αα of E that is not in E α and that, for the points :ß: α# Ðα#" Ñalready defined :

 :Á:α# :Á1Ð:Ñα# Ð‡Ñ  :Á1Ð:Ñ  #α

441 - For the “next step”, we want to define :" . Since lÒ!ß" Ñl  # ß we have so far defined fewer than - #: points α . Therefore

" - lÖ:Àααα" ×∪Ö1Ð:ÑÀ×∪Ö1Ð:ÑÀ×l# α" α α" .

- But lEElœ# cl" " " , so we can chose a limit point : " of E " with :ÂE " " so that the conditions Ð‡Ñ continue to hold for α#"    "Þ

Therefore, by transfinite recursion, we have defined distinct points :Ðα α-  Ñ in such a way that for α"-Á , 1Ð:α" Ñ Á : and 1Ð: "α Ñ Á : Þ

Let \œα- ∪Ö:α À  ×Þ By construction, \ is countably compact because every infinite set in \\ (for that matter, even every infinite set in " ) has a limit point in . But we claim that \‚\ is not pseudocompact.

To see this, consider ^ œ ÖÐ8ß 1Ð8Ñ À 8 − × © \ ‚ \Þ We claim ^ is clopen in \‚\Þ

Since Ð8ß 1Ð8ÑÑ is isolated in \ ‚ \ß ^ is a discrete open subset of \ ‚ \Þ

On the other hand, the graph of 1 œ ÖÐBß 1ÐBÑÑ À B −" × is closed in " ‚ " so that

ÖÐBß 1ÐBÑÑ À B −" × ∩ Ð\ ‚ \Ñ is closed in \ ‚ \Þ

and we claim that ÖÐBß 1ÐBÑ À B −" × ∩ Ð\ ‚ \Ñ œ ^Þ

Indeed, it is clear that

^ © ÖÐBß 1ÐBÑ À B −" × ∩ Ð\ ‚ \Ñ

and the complicated construction of the :α 's was done precisely to guarantee the reverse inclusion:

If ÐBß 1ÐBÑÑ − \ ‚ \ , then B −  for otherwise we would haveBœ:αα for some α , and then 1ÐBÑœ1Ð:ÑÂ\ by construction.

Therefore ^\‚\Þ is closed in

Therefore function 2À\‚\Ä defined by

8if ? œ Ð8ß 1Ð8ÑÑ − ^ 2Ð?Ñ œ  !?−Ð\‚\Ñ^if

continuous. But 2\‚\ is unbounded, so is not pseudocompact.

442 7. Alternate Constructions of "\

We constructed "\ by defining an order between certain compactifications of \ and showing that there must exist a largest compactification (unique up to equivalence) in this ordering. Theorem 5.7, however, shows that there are many different characterizations of "\ and some of these characterizations suggest other ways to construct "\ . .

For example, Theorem 5.7 shows that the zero sets in a Tychonoff space \ play a special role in ""\\. Without going into the details, one can construct as follows:

Let mYm be the collection of zero sets in \ . A filter in (also called a D-filter ) means a nonempty collection of nonempty zero sets such that

i) if JßJ−"#YY , then J∩J− " # , and ii) if J−YY and KªJ where K is a zero set , then K− Þ

A D\ -ultrafilter in is a maximal D -filter.

Define a set "hh\œÖ À is a D -ultrafilter in \×Þ For each :−\ , the collection hh": œÖ^À ^ is a zero set containing :× is a (trivial) D -ultrafilter, so : − \Þ The map 2Ð:Ñ œh": is a "  " map of \ into the set \ .

It turns out that \D compact iff every -ultrafilter is of the form h: for some :−\Þ Therefore the set "h\\œg iff \ is compact. Each D -ultrafilter in \ that is not of the trivial form h": is a point in \\ .

The details of putting a topology on "\ to create the largest compactification of \ are a bit tricky and we will not go into them here.

The situation is simpler in the case \œ . Since every subset of is a zero set, a “D -ultrafilter” in  is just an ordinary ultrafilter in .

Then, to be a bit more specific,

let "œÖ h À h is an ultrafilter in  × and for E©  , define clEœÖhh ÀE− ×

Give " the topology for which ÖEÀE©× cl  is a base for the open sets.

This topology make=X " into a compact # and we can embed  into " using the mapping 2Ð8Ñ œh8 ( œ the trivial ultrafilter “fixed” at 8 ). This “copy” of is dense in " , so " is a compactification of  . It can be shown that “this " ” is the largest compactification of  (and therefore equivalent to the " constructed earlier).

The free ultrafilters in "" are the points in llœ# . Since - and there - are only countably many trivial ultrafilters h8 , we conclude that there are # free ultrafilters in 

443 It turns out that the D\"" -ultrafilters in a Tychonoff space are associated in a natural way with the maximal ideals of the ring GÐ\Ñ , so it is also possible to construct " \ by putting an appropriate topology on the set

"\œÖQÀQ is a maximal ideal in GÐ\Ñ×

It turns out that if :−\ß then Q: œÖ0−GÐ\Ñ À0Ð:Ñœ!× is a (trivial) maximal ideal and the mapping 2Ð:Ñ œ Q: gives a natural way to embed \ in " \ . \ is not compact iff there are maximal ideals in GÐ\Ñ that are not of the form Q: (that is, nontrivial maximal ideals) and these are the points of "\\ .

More information about these constructions can be found in the beautifully written classic Rings of Continuous Functions (Gillman & Jerison).

In this section, we give one alternate construction of "\ in detail. It is essentially the construction used by Tychonoff, who was the first to construct "\ for arbitrary Tychonoff spaces. In his paper Uber¨ die topologische Erweiterung von Raumen¨ (Math. Annalen 102(1930), 544-561) Tychonoff also established the notation “"\ .” The construction involves a specially chosen embedding of \ into a cube.

‡ Suppose \0−GÐ\ÑM© is a Tychonoff space. For each , choose a closed 0 ‘ such ‡ that ranÐ0Ñ © M0 Þ If YY © G Ð\Ñ is a family that separates points from closed sets, then according to Theorem VI.4.10 the evaluation map given by is /À\ÄYY0−Y M0 /ÐBÑœ0ÐBÑ an embedding. In this way, every such family Y ©G‡ Ð\Ñ generates a compactification Ðcl /YY Ò\Óß / Ñ of \ . In fact, the following theorem states that every compactification of \ can be obtained by choosing the correct family Y ©G‡ Ð\Ñ .

Theorem 7.1 Every compactification of \ can be achieved using the construction in the preceding paragraph. More precisely, if ]\ is a compactification containing (with embedding 3©GÐ\Ñ), then there exists a family YY‡ such that separates points and closed sets and Ðcl /YY Ò\Óß / Ñ ¶ Ð] ß 3Ñ .

‡ µ Proof Let Y œÖ0−G Ð\ÑÀ0 can be continuously extended to 0 À] ÄM0 × . (Note that µ 0 is unique if it exists\ since any two extensions would agree on the dense set .)

The family Y separates points from closed sets:

If J\BÂJO©] is a closed set in and , then there is a closed set with BÂO∩\œJÞ By complete regularity there is a continuous function 1À] ÄÒ!ß"Ó such that 1ÐBÑ œ ! and 1lO œ "Þ Since ] is compact, 1 must be bounded and therefore 0œ1l\−GÐ\ч . Moreover, 0−Y (because 1 is the required extension). Clearly, 0ÐBÑ œ 1ÐBÑ œ ! Âcl 0ÒJ Ó © cl 1ÒOÓ œ Ö"×Þ

Therefore Ð cl /YY Ò\Óß / Ñ is a compactification of \ . µ Define 2 À ] Ä0−Y M0 by 2Ð:ÑÐ0Ñ œ 0 Ð:ÑÞ Then 2 is continuous and, for B − \ß 2ÐBÑÐ0Ñ µ  œ 0 ÐBÑ œ 0ÐBÑ œ /YYY ÐBÑÐ0ÑÞ Therefore 2Ò\Ó œ / Ò\Ó and 2 ‰ 3 œ / Þ

444 Clearly, /Y Ò\Ó œ 2Ò\Ó © 2Ò] Ó and 2Ò] Ó is compact Hausdorff, so cl /Y Ò\Ó © 2Ò] ÓÞ On the other hand, by continuity, 2Ò] Ó œ 2Ò cl \Ó © cl 2Ò\Ó œ cl /YY Ò\Ó . Therefore 2Ò] Ó œ cl / Ò\ÓÞ

Since 2 À ] Ä cl /YYY Ò\Ó is continuous and onto, Ð] ß 3Ñ ÐÐ cl / Ò\Óß / ÑÑÞ

We claim 2"" is also :

If : Á ; − ] , then there is a continuous map 1 À ] Ä Ò!ß "Ó such that 1Ð:Ñ œ ! and µµ 1Ð;Ñœ"ÞThen 0œ1l\−Y and 0 Ð:Ñœ1Ð:ÑÁ1Ð;Ñœ0Ð;ÑÞ Therefore 2Ð:ÑÐ0Ñ Á 2Ð;ÑÐ0Ñßso 2Ð:Ñ Á 2Ð;ÑÞ

Since ]/Ò\Ó2 is compact and clY is Hausdorff, is a homeomorphism and, as mentioned above, 2 ‰ 3 œ /YYY. Therefore Ð] ß 3Ñ ¶ ÐÐ cl / Ò\Óß / ÑÞ

445 Theorem 7.2 Suppose YY©©GÐ\Ñw‡ and that both Y and Y w separate points from closed sets. Then Ð cl /YYww Ò\Óß / Ñ Ð cl / YY Ò\Óß / Ñ .

For : − cl /YYw Ò\Ó , define 2Ð:Ñ − cl / Ò\Ó by 2Ð:ÑÐ0Ñ œ :Ð0Ñ œ :0 Þ ÐInformally, 2Ð:Ñ is just the result of deleting from : all the coordinates corresponding to functions in YYw .Ñ Clearly /YY œ 2 ‰ /wwwso Ð cl / YYYY Ò\Óß / Ñ Ð cl / Ò\Óß / Ñ . ñ

Corollary 7.3 A Tychonoff space \ has a largest compactification.

Proof Combining Theorems 7.1 and 7.2, we see that the largest compactification corresponds to taking Y œG‡ Ð\Ñ in the preceding construction. ñ

Of course we can do the construction (from the paragraph preceding Theorem 7.1) simply using Y œG‡ Ð\Ñ in the first place (that is what Tychonoff did) and define the resulting compactification to be "\ . We would then need to prove that it has one of the features that make it interesting for example, the Stone Extension Property. Instead, using Theorems 7.1 and 7.2, what we did was first to argue that Y œG‡ ÐGÑ produces the largest compactification of \ ; then Theorem 5.7 told us that the compactification we constructed is the same as our earlier "\ .

446 Exercises

E1. Show that the Sorgenfrey line (Example III.5.3) is not locally compact.

E2. Suppose \X is a locally compact # space that is separable and not compact. Show that the one-point compactification \‡ is metrizable.

E3. Suppose GO and are disjoint compact subsets in a locally compact Hausdorff space \YªGZªOYZ. Prove that there exist disjoint open sets and such that cl and cl are compact.

E4. a) Let O\1−GÐOÑ be a compact subspace of a Tychonoff space . Prove that for each there is an 0−GÐ\Ñ that 1œ0lO that is, every continuous real valued function on O can be extended to \ . (A subspace of \G\ with this property is said to be -embedded in . Compare Definition 6.2; for a compact since OG is compact, “ -embedded” and “ G‡ -embedded” mean the same thing.)

b) Suppose EG is a dense -embedded subspace of a Tychonoff space \0−GÐ\Ñ . If and 0ÐBÑ œ ! for some B − \ , prove that 0Ð+Ñ œ ! for some + − EÞ Hint: if 0E | is never ! , then " 0 −GÐEÑ

c) Every bounded function 0À‘ Ä has a continuous extension 0" À " Ä Þ In particular, " " the function 0Ð8Ñ œ8 can be extended. If : −"   , what is 0 Ð:Ñ ? Why does this not contradict part b) ?

E5. Prove that llœllœ#Þ"‘ " -

E6. Prove that a Tychonoff space \\ is connected iff " is connected. Is it true that \ is connected iff every compactification of \ is connected?

E7. a) Show that "‘EF ‘ has two components and .

b) Ò!ß ∞Ñ has a limit point in ""‘‘ß say in the set F . Is Ò!ß "Ñ œ F ?

E8. Let h be a free ultrafilter in .

a) Choose a point 5"− and let h œÖE©À−E× 5 cl" . Show that h is a free ultrafilter on  .

b) Using the ultrafilter hD from a), construct the space as in Exercise IX.E8. Prove that D5" is homeomorphic to ∪Ö × with the subspace topology from .

447 c) Define an equivalence relation on "BµC∪ÖB×  by if  is homeomorphic to "∪ÖC×. For B−  , let ÒBÓ be the equivalence class of B . Prove that each equivalence class satisfies lÒBÓl Ÿ - (so there must be #- different equivalence classes.)

Note: Part c) says that, in some sense, there are 2- topologically different points 5"− . By part a), each of these points 5h is associated with a free ultrafilter in that determines the topology on 5∪Ö × . Therefore there are 2- “essentially different” free ultrafilters h in .

448 Chapter X Review

Explain why each statement is true, or provide a counterexample.

1. Every Tychonoff space has a one-point compactification.

2. If \\ is Tychonoff and "" is first countable, then ±\±Ÿ- .

3. ‘ has a compactification of cardinal ##- .

4. ‘‘‘ has a compactification ]ª where ] is infinite and ] is metrizable.

5. Suppose that \B−\ is a compact Hausdorff space and that each has a metrizable neighborhood (i.e., \\ is locally metrizable ). Then is metrizable.

6. Let ‡ be the 1-point compactification of . Every subset of ‡ is Borel.

7. "  is dense in " .

8. If \œÒ 0 ß==!!  Ñ , then " \œÒ 0 ß == !!  Ó .

9. Every point in " is the from  .

10. The one-point compactification of ‘ is completely metrizable.

11. If \] and are locally compact Hausdorff spaces with homeomorphic one-point compactifications, then \] must be homeomorphic to .

12. Let 8− . All 8 -point compactifications of the Tychonoff space \ are equivalent.

13. Every subset of ‘‘ is G ‡ -embedded in .

14. If \+Ð\Ö+×Ñœ\ is compact Hausdorff and −\ , then " .

15. Every compact Hausdorff space is separable.

16. A metric space Ð\ß .Ñ has a metrizable compactification iff \ is separable.

17. ‘œY∩J for some open Y and closed J in .

449