Uncountable families of prime z-ideals in C_0(R)

0.2 MB·

by  Hung Le Pham

UNCOUNTABLE FAMILIES OF PRIME z-IDEALS IN C (R)
0
HUNGLEPHAM
8
0
0 Abstract. Denote by c= 2ℵ0 the cardinal of continuum. We construct an
2 intriguing family (Pα : α ∈c) of prime z-ideals in C0(R) with the following
properties:
Jan •• TIfif6=i∈0PPii06⊂foPri0sofmoreeia0c∈hcı0,t∈hecn. f ∈Pi forallbutfinitelymanyi∈c;
We also construct a well-ordered increasing chain, as well as a well-ordered
1 decreasing chain,of ordertypeκofprimez-idealsinC0(R)foranyordinalκ
ofcardinalityc.
]
A
R
h. 1. Introduction
t
a Let Ω be a locally compact space. In [5], we introduced the notion of pseudo-
m finite family of prime ideals as follows.
[ Definition1.1. Anindexedfamily(P ) ofprimeidealsinC (Ω)ispseudo-finite
i i∈S 0
1 if f ∈Pi for all but finitely many i∈S whenever f ∈ i∈SPi.
v
Apseudo-finitefamily(P : i∈S)ofprimeidealsinC (Ω)hasmanyinteresting
5 i S0
properties,forexample,whenS isinfinite, theunion P isagainaprimeideal
1 i∈S i
3 and any infinite subfamily of (Pi) gives rise to the same union.
S
0 Apseudo-finite family ofprime ideals(Pi : i∈S)is saidto be non-redundant if
. for every proper subset T of S, P 6= P . Non-redundancy is equivalent
1 i∈T i i∈S i
0 to either of the following ([5, Lemma 3.4]):
T T
8 (a) P 6⊂P (α6=β ∈S);
α β
0
(b) P 6⊂P for each α∈S.
: β6=α β α
v
Note that (a) is apparently weaker, whereas (b) is apparently stronger than the
i T
X non-redundancy. Thus, in this case, P cannot be written as the intersection
i∈S i
r oflessthan|S|primeidealsand|S|≤|C0(Ω)|. Furthermore,foreverypseudo-finite
a family of prime ideals, the subfamilyTconsisting of those ideals that are minimal in
the family is non-redundantandpseudo-finite andhas the same intersectionas the
original family.
The notionof pseudo-finitenesshas a connectionwith automatic continuity the-
ory. It is provedin [5] that, assuming the Continuum Hypothesis, for eachpseudo-
finite family (P : i ∈ S) of prime ideals in C (Ω) such that C (Ω)/ P = c,
i 0 0 i∈S i
there exists a homomorphism from C (Ω) into a Banach algebra whose continuity
0 (cid:12) T (cid:12)
ideal is i∈SPi. Recall that the continuity ideal is the larg(cid:12)est ideal of C0(Ω(cid:12)) on
which the homomorphism is continuous, and it is always an intersection of prime
T
ideals (see [1] for more details).
2000 Mathematics Subject Classification. Primary46J10; Secondary13C05.
Key words and phrases. Algebraofcontinuous functions,primeideal,locallycompact space.
This research is supported by a Killam Postdoctoral Fellowship and a Honorary PIMS Post-
doctoralFellowship.
1
2 HUNGLEPHAM
Suppose that Ω is metrizable and that ∂(∞)Ω♭ 6= ∅; see §2 for the definition.
Examples of such spaces include many countable locally compact spaces and all
uncountable locally compact Polish spaces. For such Ω, it is known that there
exists an infinite non-redundant pseudo-finite sequence of prime ideals in C (Ω)
0
([5]). Here, we are going to show that there exists even a non-redundant pseudo-
finite family (P :i∈c) of prime ideals in C (Ω) (Theorem 3.9). As a consequence,
i 0
assuming the Continuum Hypothesis, there exists a homomorphism from C (Ω)
0
into a Banach algebra whose continuity ideal cannot be written as intersection of
countably many prime ideals.
Note that when Ω is metrizable and ∂(∞)Ω♭ = ∅, then every non-redundant
pseudo-finite families of prime ideals in C (Ω) is finite and the continuity ideal of
0
everyhomomorphismfromC (Ω)intoaBanachalgebraistheintersectionoffinitely
0
many prime ideals. ([5])
In §4, we shall construct, for every ordinal κ of cardinality at most c, a well-
ordered decreasing chain of order type κ of prime z-filters on any uncountable
locally compact Polish space. In particular, there exists a well-ordered decreasing
chain of order type κ of prime z-filters beginning with any non-minimal prime
z-filters on R. (This was shown for κ ≤ ω2, where ω is the first uncountable
1 1
ordinal, in Theorems 8.5, 13.2 and Remark 13.2 of [3].) We also show that there
are various countable compact subspaces of R on which there is a well-ordered
decreasing chain of order type c of prime z-filters (c is identified with the smallest
ordinal of cardinality c).
In [3], it was asked whether there exists an uncountable well-ordered increasing
chain of prime z-filters on R. We shall construct in §5, for every ordinal κ of
cardinality at most c, a well-ordered increasing chain of order type κ of prime
z-filters on any uncountable locally compact Polish space. We also construct well-
ordered increasing chains of order type c of prime z-filters on various countable
compact subspaces of R.
Allthreeconstructionshaveasacommoningredientthe resultdue toSierpinski
that N can be expressed as the union of c ”almost disjoint” infinite subsets.
2. Preliminary definitions and notations
For details of the theory of the algebras of continuous functions, see [2].
Let Ω be a locally compact space; our convention is that the topological spaces
are always Hausdorff. The one-point compactification of Ω is denoted by Ω♭.
For each prime ideal P in C (Ω), either there exists a unique point p ∈ Ω such
0
that f(p)=0 (f ∈P), in which case, we say that P is supported at the point p; or
otherwise, we say that P is supported at the (point at) infinity.
It is an important fact that, for each prime ideal P in C (Ω), the set of prime
0
ideals in C (Ω) which contain P is a chain with respect to the inclusion relation.
0
Foreachfunctionf continuousonΩ,thezeroset off isdenotedbyZ(f). Define
Z[Ω] = {Z(f): f ∈C(Ω)}. For each closed subset Z ⊂ Ω, we have Z = Z(f) for
some function f ∈ C (Ω) if and only if Ω\Z is σ-compact. An ideal I of C (Ω) is
0 0
a z-ideal if g ∈I whenever g ∈C (Ω), Z(g)⊃Z(f) and f ∈I.
0
Az-filter F onΩisanon-emptypropersubsetofZ[Ω]thatisclosedunderfinite
intersection and supersets. Each z-filter F associates with the ideal
Z−1[F]={f ∈C(Ω): Z(f)∈F} of C(Ω).
PRIME IDEALS IN C0(R) 3
A z-filter P is a prime z-filter if Z ∪Z ∈/ P whenever Z ,Z ∈Z[Ω]\P.
1 2 1 2
Let P be a prime z-filter on Ω. Then we say that P is supported at a point
p∈Ω,ifp∈Z foreachZ ∈P;ifthere existsnosuchp,wesaythatP is supported
at the (point at) infinity. The supportpointof eachprimez-filterP coincides with
the support point of the prime z-ideal C (Ω)∩Z−1[P].
0
Let Ω be a compact space. Define ∂(1)Ω = ∂Ω to be the set of all limit points
of Ω. Since Ω is compact, ∂Ω is non-empty unless Ω is finite. We then define
inductively a non-increasing sequence ∂(n)Ω:n∈N of compact subsets of Ω by
setting ∂(n+1)Ω = ∂ ∂(n)Ω for each n ∈ N. Set ∂(∞)Ω = ∞ ∂(n)Ω. By the
(cid:0) (cid:1) n=1
compactness, either ∂(∞)Ω is non-empty or ∂(l)Ω is empty for some l∈N.
(cid:0) (cid:1) T
A Polish space is a separable completely metrizable space. Every separable
metrizable locally compact space is a Polish space.
3. Pseudo-finite families of prime ideals and prime z-filters
Let Ω be a locally compact space. First, we shall make a connection between
pseudo-finite families of prime ideals andpseudo-finite families of prime z-idealsin
C (Ω). For each closed subset E of Ω, we define the z-ideal
0
K ={f ∈C (Ω): E ⊂Z(f)}.
E 0
The following is [4, 3.3 and 3.4], we shall give here a combined proof.
Lemma 3.1. [4] Let I be an ideal in C (Ω). Then
0
Iz = {K : E is closed in Ω and K ⊂I}
E E
is the largest z-ideal con[tained in I. If I is a prime ideal then so is Iz.
Proof. ItiseasytoseethatIz containseveryz-idealcontainedinI. Since thesum
of two z-ideals is again a z-ideal, we see that
Iz = {K : E is closed in Ω and K ⊂I};
E E
where the sum is algebXraic. Thus Iz is the largest z-ideal contained in I.
Now,suppose that I is prime. Letf ,f ∈C (Ω)\Iz. Then there existsg ,g ∈
1 2 0 1 2
C (Ω)\I such that Z(g ) ⊃ Z(f ). Then Z(g g ) ⊃ Z(f f ). The primeness of I
0 i i 1 2 1 2
implies that g g ∈/ I. So f f ∈/ Iz. (cid:3)
1 2 1 2
The following strengthens the implication (a)⇒(c) of [5, Lemma 8.4].
Proposition 3.2. Let Ω be a locally compact space. Let (P : i∈S) be an infinite
i
non-redundant pseudo-finite family of prime ideals in C (Ω). Then P = P is
0 i∈S i
a prime z-ideal, and (Pz : i∈S) is a non-redundant pseudo-finite family of prime
i
z-ideals whose union is P such that Pz ⊂P (i∈S). S
i i
Proof. We shall need another theorem of [4] which say that the sum of two non-
comparable prime ideals in C (Ω) is indeed a prime z-ideal ([4, 3.2]).
0
We know that P must be a prime ideal. Assume toward a contradiction that P
is not a prime z-ideal. Choose α 6= α ∈ S arbitrary. Then P +P is a prime
1 2 α1 α2
z-ideal. Suppose that we already have distinct indices α ,...,α ∈ S such that
1 n
n n
P is a prime z-ideal. Then P 6= P , and so we can find α ∈ S
i=1 αi i=1 αi n+1
P P
4 HUNGLEPHAM
n
suchthat P 6⊂ P . The induction can be continued. However,this gives
αn+1 i=1 αi
a contradiction since then
P
∞ ∞ n
P = P = P
αn αi
n=1 n=1i=1
[ [ X
is a z-ideal. Hence, P is a prime z-ideal.
We claim that (Pz : i ∈ S) is a pseudo-finite family with union P. Indeed,
i
assumetowardacontradictionthatthereexistsf ∈P anddistinctα ∈S (n∈N)
n
such that f ∈/ Pz (n ∈ N). For each n, we can then find f ∈/ P such that
αn n αn
Z(f ) ⊃ Z(f); we can further assume that 0 ≤ f ≤ 2−n. Define f = ∞ f .
n n ∗ n=1 n
Then we see that f ≤f so f ∈/ P (n ∈N), and that Z(f )⊃Z(f) so f ∈P.
n ∗ ∗ αn ∗ P∗
This is a contradiction to the pseudo-finiteness of (P : i∈S).
i
It remains to prove the non-redundancy of (Pz : i ∈ S). So, assume that
i
Pz ⊂Pz for some α6=β ∈S. Then Pz is containedin both P and P , and so P
α β α α β α
and P are in a chain. This contradicts the non-redundancy of (P : i∈S). (cid:3)
β i
Conversely, it is obvious that if (Q : i ∈ S) is a pseudo-finite family of prime
i
(z-)ideals and P is a prime ideal containing Q and contained in Q (i∈S)
i i α∈S α
then (P : i∈S) is a pseudo-finite family of prime ideals.
i
S
We define a similar notion of pseudo-finite families of prime z-filters.
Definition 3.3. An indexed family (P ) of prime z-filters Ω is pseudo-finite if
i i∈S
Z ∈P for all but finitely many i∈S whenever Z ∈ P .
i i∈S i
A pseudo-finite family of primez-filters(Pi : i∈SS)is saidto be non-redundant
ifforeverypropersubsetT ofS, P 6= P . Similarto [5, Lemma3.4]we
i∈T i i∈S i
have the following.
T T
Lemma 3.4. Let (P : α ∈ S) be a pseudo-finite family of prime z-filters on Ω.
α
Then the following are equivalent:
(a) (P ) is non-redundant;
α
(b) P 6⊂P (α6=β ∈S);
α β
(c) P 6⊂P for each α∈S.
β6=α β α
Proof. OTbviously, (c)⇒(a)⇒(b).
We now prove (b)⇒(c). Fix α ∈ S. By condition (b), P 6⊂ P (β ∈ S \{α}).
β α
Choose Z ∈ P \P for some β ∈ S \{α}. Then, by the pseudo-finiteness,
0 β0 α 0
we have Z ∈ P for all but finitely many β ∈ S. Let β ,...,β be those indices
0 β 1 n
β ∈S\{α} such that Z ∈/ P . For each1≤k ≤n, choose Z ∈P \P , andset
0 β k βk α
n
Z = Z . Then Z ∈ P (β ∈ S \{α}), but Z ∈/ P , by the primeness of P .
k=0 k β α α
Thus (c) holds. (cid:3)
S
The following definition and proposition are adapted from [5].
Definition 3.5. LetΩbealocallycompactspace,andletS beanon-emptyindex
set. Let F be a z-filter onΩ, and let (Z : α∈S)be a sequence ofzero sets onΩ.
α
Then F is extendible with respect to (Z : α∈S) if both the following conditions
α
hold:
(a) Z ∈/ F, and Z ∪Z ∈F (α6=β ∈S);
α α β
(b) for each Z ∈Z[Ω], if Z ∪Z ∈F for some α ∈S, then Z ∪Z ∈ F for
α0 0 α
all except finitely many α∈S.
PRIME IDEALS IN C0(R) 5
Proposition 3.6. Let Ω be a locally compact space. Suppose that there exist a
z-filter F and a family (Z : α ∈ S) in Z[Ω] such that F is extendible with
α
respect to (Z : α∈S). Then there exists a pseudo-finite family of prime z-filters
α
(P : α∈S) such that Z ∈ P \P for each α∈S.
α α γ6=α γ α
Proof. We see that the unionTof a chain of z-filters, each of which contains F and
is extendible with respect to (Z ), is also extendible with respect to (Z ). Thus,
α α
by Zorn’s lemma, we can suppose that F is a maximal one among those z-filters.
For each α, set F ={Z ∈Z[Ω]: Z∪Z ∈F}, and set P = F . By the
α α α∈S α
extensibility of F, we see that whenever Z ∈P then Z ∈F for all except finitely
α
S
many α∈S. Thus, in particular, the set P is actually a z-filter.
Claim 1: For each Z ∈ Z[Ω] \ P, we have {Z ∈Z[Ω]: Z∪Z ∈F} = F.
0 0
Indeed, we see that Z ∈/ G = {Z ∈Z[Ω]: Z∪Z ∈F} (α ∈ S); for otherwise,
α 0
Z would be in P. It then follows easily that G is extendible with respect to (Z ).
0 α
This and the maximality of F imply the claim.
Claim 2: P is a prime z-filter. We have to prove that, whenever Z ,Z ∈ Z[Ω]
1 2
are such that Z ∪Z ∈ P, but Z ∈/ P, then Z ∈ P. Indeed, let α ∈ S be such
1 2 1 2 0
that Z ∪Z ∪Z ∈F. Then Z ∪Z ∈F, by the first claim, and so Z ∈F .
α0 1 2 α0 2 2 α0
Now, for each α∈S, define
D ={Z ∪Z : Z ∈Z[Ω]\P}.
α α
Then,by Claim2,the setD isclosedunder finite union. Obviously,D ∩F =∅.
α α α
Thus, there exists a prime z-filter P containing F such that D ∩P =∅.
α α α α
We see that F ⊂P ⊂P and Z ∈/ P (α∈S). The result then follows. (cid:3)
α α α α
We now define a “prototype” space Ξ. Denote by ∞ the point adjoined to N to
obtain its one-point compactification N♭. The product space (N♭)N is a compact
metrizable space. Define Ξ to be the compact subset of (N♭)N consisting of all
elements (n ,n ,...) with the property that there exists k ∈ N such that n ≥ k
1 2 i
(1≤i≤k) and such that n =∞ (i>k). The convention is that ∞>n (n∈N).
i
Lemma 3.7. [5, Lemma 9.2] Let Ω be a locally compact metrizable space. Suppose
thatthereexistsapointp∈∂(∞)(Ω♭). Thenthereexistsahomeomorphicembedding
ι of Ξ onto a closed subset of Ω♭ such that ι(∞,∞,...)=p.
AkeytoourconstructionistheresultduetoSierpinskithatthereexistsafamily
{E : α∈c} of infinite subsets of N satisfying the following properties:
α
(i) N= E , and
α∈c α
(ii) E ∩E is finite for each α6=β ∈c.
α β
S
We sketch the nice construction of such family as follows (cf. [7]): The set N is
isomorphic to
∞
C = {f :{1,...,n}→{1,2}}.
n=1
[
For each f :N→{1,2}, define
C ={the restrictions of f to {1,...,n}: (n∈N)}.
f
We see that C = C and that C ∩C is finite for each f 6=g. We can
f:N→{1,2} f f g
thenmapbackfromC toN. Inspectingtheconstruction,weseethat{E : α∈c}
S α
enjoys the following property:
(i’) The cardinality of {α∈c: n∈E } is c for each n∈N.
α
6 HUNGLEPHAM
Lemma 3.8. There exists a non-redundant pseudo-finite family {Q : α∈c} of
α
prime z-filters on Ξ such that each z-filter is supported at the point (∞,∞,...).
Proof. Let {E : α∈c} be the family of infinite subsets of N as in the previous
α
paragraph. For each α∈c, define
N ={(j ,j ,...)∈Ξ: j =∞ (n∈E )}.
α 1 2 n α
Let F to be the z-filter generatedby allN ∪N (α,β ∈c, α6=β). We claim that
α β
F is extendible with respect to (N : α∈c); the proof will then be completed by
α
applying Lemma 3.6.
Obviously,N ∪N ∈F (α6=β). Weclaim thatN ∈/ F (α∈c). Indeed,assume
α β α
thecontrary. Thenthereexistγ ,...,γ ∈c\{α}suchthatN ⊃ m N . Since
1 m α i=1 γi
m
E is infinite whereas each E ∩E is finite, there exists l ∈ E \ E . We
α m α γi αT i=1 γi
see that (j )∈ N \N where j =l and j =∞ (i6=l); a contradiction.
Finally,isuppois=e1thaγit N ∈α Z[Ξ] sulch that Ni∪N ∈ F for someSα ∈ c. Then,
α
there exist γ ,.T..,γ ∈c\{α} such that
1 m
m m
N ∪N ⊃ N , and so N ⊃ N \N .
α γi γi α
i=1 i=1
\ \
As above, there exists l ∈ E \ m E . We can then choose γ ,...,γ ∈ c
α i=1 γi m+1 n
such that
S n
{1,...,l}⊂ E .
γi
i=1
[
n n
Weclaim thatN ⊃ N . Indeed,let(j )∈ N . Thenj =···=j =∞.
i=1 γi i i=1 γi 1 l
We see that there exists k ≥ l such that j ≥ k (1 ≤ i ≤ k) and j = ∞ (i > k).
i i
T T
For each r ∈ N, set j(r) = j (i 6= l) and set j(r) = k +r. Then, we see that
i i l
(j(r)) ∈ m N \N ⊂N and lim (j(r))= (j ). Thus (j )∈ N. Hence, for each
i i=1 γi α r i i i
β ∈c\{γ ,...,γ }, we have N ∪N ∈F. (cid:3)
1 n β
T
Theorem 3.9. Let Ω be a locally compact metrizable space. Suppose that p ∈
∂(∞)(Ω♭). Then there exists a non-redundant pseudo-finite family (P : α ∈ c) of
α
prime z-filters on Ω, each z-filter is supported at p.
Moreover, by setting P =C (Ω)∩Z−1[P ], we obtain a non-redundant pseudo-
α 0 α
finite family of prime z-ideals in C (Ω), each ideal is supported at p, such that
0
C (Ω) P =c.
0 α
(cid:12)(cid:12) (cid:30)α\∈c (cid:12)(cid:12)
(cid:12) (cid:12)
Proof. In this proof, we shall(cid:12)identify Ξ wit(cid:12)h a closed subset of Ω♭ such that
(cid:12) (cid:12)
(∞,∞,...) is identified with p; in the case where p ∈ Ω, we can further assume
that Ξ⊂Ω (cf. Lemmas 3.7).
Let (Q : α∈c) be the family of prime z-filters on Ξ as constructed in Lemma
α
3.8. For each α∈c, set
P ={Z ∈Z[Ω]: (Z∪{p})∩Ξ∈Q }.
α α
Note thateveryclosedsubsetofΞis inZ[Ξ], sowe cansee thateachP is a prime
α
z-filteronΩ. Thepseudo-finitenessof(P : α∈c)andof(P : α∈c)thenfollows
α α
from that of (Q ). The cardinality condition follows from the fact that |C(Ξ)|=c.
α
PRIME IDEALS IN C0(R) 7
By the non-redundancy of (Q : α∈c), for each α∈c, there exists
α
N ∈ Q \Q .
α β α
β6=α
\
Bythe Urylson’slemma, wecanfindZ ∈Z[Ω]suchthat(Z ∪{p})∩Ξ=N ; we
α α α
can even require Ω\Z to be σ-compact so that Z =Z(f ) for some f ∈C (Ω).
α α α α 0
Thus we see that
Z ∈ P \P and f ∈ P \P .
α β α α β α
β∈c,β6=α β∈c,β6=α
\ \
Finally, we shall prove that each P (and hence each P ) is supported at p
α α
(α ∈ c). Indeed, in the case where p is the point at infinity of Ω, for each x ∈ Ω,
there exists N ∈Q such that x∈/ N. We can then find Z ∈Z[Ω] such that x∈/ Z
α
andthat(Z∪{p})∩Ξ=N. Thus Z ∈P andx∈/ Z. So P is supportatinfinity.
α α
Ontheotherhand,inthecasewherep∈Ω,letZ ∈P bearbitrary. ThenZ∩Ξis
α
closedinΞ,andsoitisinZ[Ξ]. Since{p}∈Z[Ξ]\Q ,wededucethatZ∩Ξ∈Q .
α α
Hence, p∈Z, and thus P is supported at p. (cid:3)
α
Corollary 3.10. Let p∈R♭. There exists a family (P : α∈c) of prime z-ideals
α
in C (R) with the following properties:
0
• If f ∈P for some α ∈c, then f ∈P for all but finitely many α∈c;
α0 0 α
• P 6⊂P for each α ∈c;
• eaαc6=hαP0 αis supαp0orted at p. 0 (cid:3)
α
T
There are many countable compact metrizable spaces Ω with ∂(∞)Ω 6= ∅. We
note as a specific example the following countable compact subset of [0,1]:
k
∆={0}∪ 2−ni : k, n ,n ,...,n ∈N and k ≤n <···<n .
1 2 k 1 k
( )
i=1
X
Corollary 3.11. There exists a family (P : α∈c) of non-modular prime z-ideals
α
in C (∆\{0}) with the following properties:
0
• If f ∈P for some α ∈c, then f ∈P for all but finitely many α∈c;
α0 0 α
• P 6⊂P for each α ∈c. (cid:3)
α6=α0 α α0 0
4. WellT-ordered decreasing chains of prime ideals and prime z-filters
Let Ω be a metrizable locally compact space. If ∂(n)Ω♭ = ∅ for some n ∈ N,
then itcan be seen thateverychainof primez-idealsin C (Ω) orprime z-filterson
0
Ω has length at most n. Hence, in this section we shall suppose that ∂(∞)Ω♭ 6=∅.
Inthefollowing,Ξisthecompactsubsetof(N♭)Ndefinedintheprevioussection.
Also, our convention is that max∅ is smaller and min∅ is bigger than everything,
andthat N is the whole space (i.e. Ξ inthe nextlemma) and N =∅.
α∈∅ α α∈∅ α
Lemma T4.1. There exists a family (Nα)α∈c of zero sets on Ξ satisfSying that, for
every γ ∈c and disjoint finite subsets F and G of c, we can find a finite subset H
of c with the properties that γ ≤minH and that
N \ N ⊃ N .
β α β
β\∈G (cid:16)α[∈F (cid:17) β∈\G∪H
Proof. Recall from the previous section that there exists a family (E : α∈ κ) of
α
infinite subsets of N satisfying:
8 HUNGLEPHAM
(a) E ∩E is finite for each α6=β ∈κ, and
α β
(b) the cardinality of {α∈c: n∈E } is c (n∈N).
α
Similar to Lemma 3.8, we define, for each α∈c,
N ={(j ,j ,...)∈Ξ: j =∞ (n∈E )}.
α 1 2 n α
LetGbeafinitesubsetofκandletα,γ ∈c. Then,thereexistsl∈E \ E .
α β∈G β
Sincethecardinalityof{β : β <γ}islessthanc,by(b)above,wecanfindafinite
S
subset H of c such that γ ≤minH and that
{1,...,l}⊂ E .
β
β∈G∪H
[
Then, similar to Lemma 3.8, we see that
N \N ⊃ N .
β α β
β∈G β∈G∪H
\ \
The general case follows by induction. (cid:3)
Theorem 4.2. Let Ω be a metrizable locally compact space, and let p ∈ ∂(∞)Ω♭.
Then there exists a well-ordered decreasing chain (Q : α ∈ c) of prime z-filters
α
on Ω each supported at p.
Furthermore, by setting Q = C (Ω)∩Z−1[Q ], we obtain a well-ordered de-
α 0 α
creasing chain (Q : α∈c) of prime z-ideals in C (Ω) each supported at p.
α 0
Proof. Similar to Theorem 3.9, we shall identify Ξ with a closed subset of Ω♭ such
that(∞,∞,...)isidentifiedwithp;inthecasewherep∈Ω,wecanfurtherassume
that Ξ⊂Ω (cf. Lemmas 3.7).
Let(N : α∈c)bethefamilyofzerosetsonΞasconstructedinLemma4.1. For
α
eachα∈c,chooseZ =Z(f )forsomef ∈C (Ω)suchthat(Z ∪{p})∩Ξ=N .
α α α 0 α α
Also, define
n
F = Z ∈Z[Ω]: Z∪{p}⊃ N for some α≤β ,...,β ∈c .
α βi 1 n
( )
i=1
\
Then (F ) is a decreasing c-sequence of z-filters on Ω; Z ∈ F but Z ∈/ F
α α α α β
(α<β ∈c).
Set
D ={Z ∈Z[Ω]: (Z∪{p})∩Ξ={p}}.
∗
Then D is closed under taking finite union. Also, since D ∩F = ∅, there exists
∗ ∗ 0
a prime z-filter Q on Ω containing F such that Q ∩D =∅. Let γ ∈c. Suppose
0 0 0 ∗
that we have already constructed a well-ordered decreasing chain (Q : α < γ)
α
of prime z-filters on Ω such that F ⊂ Q (α < γ). If γ is a limit ordinal, set
α α
Q = Q . Consider now the case where γ =α+1 for some α. Set
γ α<γ α
T Dγ ={Z∪Zα : Z ∈Z[Ω]\Qα}.
Then D is closed under taking finite union. Also, we have F ∩D = ∅; since
γ γ γ
otherwise,thereexistZ ∈Z[Ω]\Q andafinite subsetGofcsuchthatγ ≤minG
α
and that
Z∪Z ∪{p}⊃ N which implies that Z∪{p}⊃ N \N ⊃ N
α β β α β
β∈G β∈G β∈H
\ \ \
PRIME IDEALS IN C0(R) 9
for some finite subset H of c with γ ≤ minH, by Lemma 4.1, or Z ∈ F ⊂ Q a
γ α
contradiction. Therefore, there exists a prime z-filter Q such that F ⊂ Q and
γ γ γ
Q ∩D = ∅. We see that, in this case, Q ( Q and Z ∈/ Q . Thus, in both
γ γ γ α α γ
cases, the construction can be continued inductively.
Setting Q = C (Ω)∩Z−1[Q ]. Then f ∈/ Q for f ∈ C (Ω) such that Z(f)∈
α 0 α ∗ 0 ∗ 0
D , and, for each γ = α+1 ∈ c, we have f ∈ Q \Q . It follows that the chain
∗ α α γ
(Q : α∈c) is decreasing.
α
The statement on support point follows from the fact that N ={p}. (cid:3)
α∈c α
It was proved in [3, Theorem 13.2] that starting from anyTnon-minimal prime
z-filter containing a countable zero set on R there exists a well-ordered decreasing
fullω -sequenceofprimez-filterssuchthateachprimez-filtercontainsacountable
1
zeroset. However,besidesthatω <cintheabsenceoftheContinuumHypothesis,
1
theunionofthosecountablezerosetsarenotcountable,andthusthatω -sequence
1
says nothing about uncountable chains of prime z-filters on countable spaces.
Corollary 4.3. Let ∆ be any countable compact subset of R such that ∂(∞)∆6=∅.
There exists a well-ordered decreasing chain of order type c of prime z-filters on R
such that each prime z-filter contains ∆. (cid:3)
We now look for longer chains. We shall need to restrict to uncountable locally
compact Polish spaces. Note that for any well-ordered decreasing chain of order
type κ of prime z-filters on Ω or prime ideals in C (Ω), where Ω is in addition
0
σ-compact, κ must have cardinality at most c.
Lemma 4.4. Let κ be an ordinal of cardinality c. There exists a family (N )
α α∈κ
of zero sets on (N♭)N such that for every disjoint finite subsets F and G of κ, we
have
N \ N = N .
α β α
α\∈F (cid:16)β[∈G (cid:17) α\∈F
Proof. Similar to (but simpler than) that of Lemma 4.1. (cid:3)
Theorem 4.5. Let Ω be an uncountable locally compact Polish space. Let κ be an
ordinal of cardinality c. Then there exists a well-ordered decreasing chain (Q :
α
α∈κ) of prime z-filters on Ω.
Furthermore, by setting Q = C (Ω)∩Z−1[Q ], we obtain a well-ordered de-
α 0 α
creasing chain (Q : α∈κ) of prime z-ideals in C (Ω).
α 0
Proof. Every uncountable Polish space contains a closed subsets homeomorphic to
the Cantor space {0,1}N, which in turn contains a copy of (N♭)N. Thus, we shall
identify (N♭)N with a closed subset of X of Ω where (∞,∞,...) is identified with
some point p. Let (N : α∈ κ) be the family of zero sets on X as constructed in
α
Lemma 4.4. Define
n
F = Z ∈Z[Ω]: Z ⊃ N for some α≤β ,...,β ∈κ .
α βi 1 n
( )
i=1
\
The remainingof the proofis similar to that ofTheorem 4.2, but applying Lemma
4.4 instead of Lemma 4.1. (cid:3)
Corollary 4.6. Let κ be any ordinal of cardinality c. Then:
10 HUNGLEPHAM
(i) There exists a well-ordered decreasing chain of order type κ of prime z-
filters on R starting from any non-minimal prime z-filter.
(ii) There exists a well-ordered decreasing chain of order type κ of prime z-
ideals in C (R) starting from any non-minimal prime z-ideals.
0
Proof. (i) followsfrom the theoremand [3, Theorem12.8],and(ii) follows from(i)
and the fact that every zero set on R is the zero set of a function in C (R). (cid:3)
0
5. Well-ordered increasing chains of prime ideals and prime z-filters
Let Ω be a metrizable locally compact space. Similar to the previous section
we shall only consider the case where ∂(∞)Ω♭ 6= ∅. Recall that there are many
countable compact space satisfying this condition. First we shall prove a general
construction.
Definition 5.1. Let κ be any ordinal, and let (Z : α ∈ κ) be a family of zero
α
sets on Ω. A zero set Z is said to have property (A) (with respect to the family
(Z : α ∈ κ)) if for every (possibly empty) finite subset F of κ and every β ∈ κ
α
with maxF <β then
Z∩ Z 6⊂Z .
α β
α∈F
\
AzerosetZ issaidtohaveproperty(B)(withrespecttothefamily(Z : α∈κ))
α
if whenever Z = n Z for some Z ,...,Z ∈ Z[Ω] then there exists 1 ≤ k ≤ n
i=1 i 1 n
such that Z has property (A).
k
S
Lemma 5.2. Let κ be any ordinal, and let (Z : α ∈ κ) be a family of zero sets
α
on Ω. Suppose that F is a z-filter on Ω such that every element of F has property
(B) with respect to (Z : α∈κ). Then there exists a well-ordered increasing chain
α
(Q : α ∈ κ) of prime z-filters containing F such that Z ∈/ Q but Z ∈ Q
α α α α β
(α<β ∈κ).
Proof. Let D be the collection of all zero sets not having property (B). Then ob-
viously D is closed under finite union and (Z : α ∈ κ) ⊂ D. Since F ∩D = ∅,
α
there exists a prime z-filter Q such that F ⊂ Q and Q ∩ D = ∅. We then
0 0 0
define Q to be the z-filter generated by Q and {Z :γ <α}. It follows that Q
α 0 γ α
is a prime z-filter, Q ⊂ Q and Z ∈ Q (α < β ∈ κ). We need to show that
α β α β
Z ∈/ Q (α ∈ κ) (and thus (Q ) is increasing). Assume towards a contradiction
α α α
that Z ∈ Q for some α ∈ κ. Then, there exist N ∈ Q and a finite subset F of
α α 0
{γ :γ <α} such that
Z ⊃N ∩ Z .
α γ
γ∈F
\
This implies that N ∈D a contradiction. (cid:3)
Lemma 5.3. There exists a family (Nα)α∈c of zero sets on Ξ satisfying that Ξ has
property (B) with respect to (N : α∈c).
α
Proof. Let (E : α∈c) and (N : α∈c) be defined as in Lemma 4.1.
α α
We shallprovea little strongerstatement. Assume towardsacontradictionthat
there exists Z ,...,Z ∈Z[Ξ] such that Ξ= n Z and that, for each 1≤i≤n,
1 n i=1 i
there exist finite subsets F and G of κ with maxF <minG such that
i i i i
S
Z ∩ N ⊂ N .
i α β
α\∈Fi β[∈Gi