Zcitrchr. 1. W. L w i k und Grundlugen d. Math. Bd. 17, S. 377-384 (1971)

THE COFINAL CHARACTER OF UNIFORM SPACES AND ORDERED FIELDS

by PAUL HAFNER and G U E ~ O WZOLA in Zurich (Switzerland)

Introduction The first idea in writing this paper was to generalize some results of HAUSCHILD [7]

to a special class of uniform spaces. Later, we found that a certain number of our results was already available in the literature. It seems, however, still worthwile to present an exposition of these ideas.

KALISCH [9] proved that every uniform space X is a generalized metric space in the sense that the distance function takes its values in a partially ordered abelian group. Lf we impose on X the restrictive condition that there must be a nested base for the uniformity, the above group can be chosen to be linearly ordered (COHEN/ G O ~ M A N [4]). By the total order (under inclusion) of the uniformity base it is possible to associate a regular ordinal number x ( X ) to the space X ([2], [3], [4]; for ordered fields, this has been remarked also by SIKORSEI [12]). The completion of such spaces may be achieved by means of CAucm-sequences of type x(X) ([3], [7], [ I l l ) . Further developments may be found in [ll], [13], [14].

$0 1 and 2 of the present paper give a general exposition of the situation found in the study of ordered fields - yielding the generalization suggested by HAUSCHILD ([7], p. 65). $ 3 offers a new characterization of ordered fields which are dense in their real closure.

§ 1. The Cofinal Character We introduce a list of concepts which we are going to apply and justify in the

following propositions: it is well known what is meant by the cofinal and coinitial characters of a totally ordered set (see for instance HAUSDORFF [S] , pp. 129-133); these are regular initial ordinals. The character of an element x in an ordered set is defined to be the pair [a , PI, where a denotes the cofinal character of the subset of all elements smaller than x, and P denotes the coinitial character of the subset of all elements greater than x ([S], p. 143).

I n the following definitions, S is an arbitrary set and ( X , U) is a uniform space. Sequences are understood to be functions on an ordinal (not necessarily denumerable).

Defini t ions. a) The cofinal character of a filter 3 in S is the cofinal character of a nested base if there is any, and zero otherwise.

b) The cofinal character of a (transfinite) sequence F is the cofinal character of its domain.

c) The cofinal chmcter of a uniform space ( X , U) is defined to be the cofinal character of the uniformity filter U and denoted by X(X).

d) In (x, U) a CAUCHY-SeqUenCe is termed proper iff it is not eventually constant; a CaucHY-filter is proper iff its character is greater than 1 .

26 Ztschr. f . math. Log&

378 PAUL HAFNER AND GUERINO MAZZOLA

e) The sequence character of ( X , U), denoted by XF(X), is the smallest ordinal figuring as character of a proper CAUCHY-SeqUenCe in x. If there are no such se- quences, we define XF(X) = 0. Similarly, the filter character of ( X , U), denoted by XF(X), is the smallest ordinal appearing as character of a proper CAucHY-filter if there is any, zero otherwise.

The following lemma will simplify considerably the confusion of characters made above :

Lemma. If ( X , U) is a T, uniform space with a non discrete topology and a uni- formity base totally ordered by inclusion, then the following statements hold:

X(X) = X F ( X ) = X,(X). Every proper CAumy-sequence and every proper Cmcrry-filter are precisely of the character X(X) . Every accumulation point of X is limit of proper Caucwz-sequences and of proper CAucHY-filters. If X is ordered such that t7w order topology is the uniform topology, then the cha?.acter [a, ,4] of a point is always either [l , X(X)] or [X ( X ) , 13 or [l , 11 or [X (XI 9 X (X)l. If X is a topological group, then every point in X is limit of proper CAUCHY- sequences and of proper CAUCHY-filterS. If in addition the topological group X is an ordered group such that the two topo- logies coincide, then the character of all points is [X(X), X(X)].

Proof. To begin, we demonstrate the equality X(X) = XF(X). There is a base 93 of the uniformity U which is well-ordered by inclusion: % = {B, I L < I}. Since X is not discrete, there is an accumulation point xo in X. Clearly, the set {B, [xO] I L < I ] , where B, [zO] = {y I ( 5 , y) E B,}, is a base of the neighbowhood filter of xo. From each B,[xo] we may choose a y, + xo and define a proper CAucHY-sequence of co- final character at most X ( X ) . Hence XF ( X ) 2 X ( X ) .

In order to verify XF(X) 2 X (X) , we construct inductively an increasing function ~ 1 : XF(X) + X(X). It will turn out that XF(X) is cofinal with X(X) and therefore XF(X) 2 X(X), because X(X) is regular.

Let {a, I Y < XF(X)) be a proper CaucHY-sequence and assume that v lyD, yo < XF(X) is already defined and increasing. yo is not cofinal with X(X) by the inequality XF(X) 5 X(X). Now, we define y(v0) to be the first L < X(X) satisfying the conditions L > y ( y ) for all Y < yo and B, does not contain every (a,, a,) for Y , ,u 2 yo . The second one is not void by the choice of our sequence and by the T, property of X .

It remains to be shown that the image of y is cofinal with X ( X ) , but this is straight- forward: consider B, for I < X(X). Since (a, 1 Y < Xp(X)} is CAUCHY, there is a y1 such that (a,, a,) is in B, for all Y, p 2 Y ~ . Hence y(vl) must be greater than L

by the construction of y . This shows XF(X) 2 X(X) and thus equality. In fact, our proof settled (iii) for sequences. Obviously, XF(X) = X,(X). So far, we have proved (i) and (iii). (ii) is evident, for one can choose from each

proper CaucHY-sequence a subsequence which is Cauchy and of the character X ( X ) .

THE COFINAL CHARACTER OF UNlPORM SPACES AND ORDERED FIELDS 379

(v) is consequence of the translation property of neighbourhoods in topological groups.

(iv) is immediate from the definition of [a, ,191 and (ii). Finally, (vi) is consequence of the continuity of the inversion operation on a topological group.

Remark . It is essential t o assume X to have a non discrete topology, for there are uniform spaces with discrete topology and nested uniformity base of arbitrarily high cofinal character. See for instance exercise C chapter6 in KELLEY [lo]. (If the uniform topology of X is discrete, proper CAucHy-sequences exist iff X is not complete.)

As more or less direct consequences of the above lemma, some rather interesting theorems can be proved. They are all suitable generalisations of propositions about ordered fields which appear in HAUSCHILD [7]. However, one of €€AUSCHILD’S pro- positions will be disproved by a counterexample. - The results deal with the CANTOR completion of topological groups with nested uniformity base and should provide it simpler treatment of these structures than the filter completion. To begin, we state the

Theorem 1. If Y is a dense subspace of a uniform T, space X with the cofinality character X(X) 2 1 , then Y , with the relativized uniformity, has the same character. In particular, the cofinal characters of a T, space Y with uniformity character X ( Y ) 2 1 ~ n d its completion coincide.

Proof . Evidently, the character of the relativized uniformity is a t most X(X). To show the equality, we are going to construct a nested base of X that maintains its character through the relativization.

Let B be a nested base of the uniformity. Then the family of the closures of the base elementa is also a base. Since the interior of a uniformity element is a uniformity element and since the space X is T,, we may find a subfamily of the closed base such that every successor Blfl of a member B, is properly contained in the interior of B,. Hence the relativized base will maintain the character by the density of Y x Y in X x X. Finally, since a nested base yields a nested base of the completion, the theorem is proved.

§ 2. Completion It is well known that a topological group which is T, and satisfies the first axiom

of countability can be completed by countable CAucHY-sequences; see VAK DANTZIG [5]. The generalisation of this fact to groups of arbitrary character is obvious, more precisely: Every T, topological group G with a cofinal character x(G) 2 1 and with the condition that the inverse of a CAucHY-sequence is again a CAncHY-sequence can be completed by X (G) CAucHY-sequences, that is sequences with the domain X (G) . - With abelian groups and especially with rings the last condition becomes trivial. Thus, every T, topological ring R of a character X(R) 2 1 can be completed by CAucHY-sequences of the length X (R) . I f the ring R is a topological skewfield k, then, under the condition that the multiplicative inverse of a CAucHu-sequence not converging to zero is again a CAUCHY-Sequence, the completion by CAuc-cHY-sequences yields a complete skewfield. Especially, these results hold for all valuated fields.

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380 PAUL EAFNER AND QUERINO MAZZOLA

Let a topological group G fullfill the conditions for completion noted above. One may ask: when does the CANTOR completion process lead to the completion of the group G if we vary the length of the Caucmr-sequences? The answer is, if G is not yet complete, the

Theorem 2. The CANTOR completion process with Caumy-sequences leads to the completion of a compktable group G of character X(G) 2 1 iff X(G) is the character of the CAUCHY-SeqUenCes used in the process.

Proof. If we use Caucmr-sequences for the CANTOR process which are not of the character %(a), then by the lemma they must all be eventually constant and the CANTOR process leads to the group again.

If we use CauaHY-sequences of the character X (a), then we may assign to each aequence a subsequence of the length X (G) which is cofinal with the sequence. This function clearly defines an isomorphism from the group obtained by the CANTOR procesa onto the completion of G.

H ~ U S ~ D states in [7] that if in an ordered field there is a proper CAUCHY- sequence of length a and a proper CaumY-sequence of length ,9, a < ,9, then a is cofinal with /I. This does not seem to be true. As a counterexample one can take a = co, * coo and ,9 = a + coo. Both a and ,9 are of the character coo but since col is not countable no increasing function from a into ,9 is cofinal in ,9. If we define two CAUCHY-Sequences, one on a and the other on B , both proper in Q , we are in contradiction to the statement of HAUSUHILD.

H~USCHILD'S proposition5 (and the idea of proof) becomes correct if a is the smallest regular initial ordinal for which proper CAUCHY sequences exist ; of course, no further result of HAUSCHILD is affected by this slip. This makes clear that it is reasonable to confine the interest from the beginning to a-sequences, a a regular initial ordinal.

It should be mentioned that theorem 2 is valid for completable rings and skew- fields too, if they are of character X 2 1 .

Finally, we shall prove the useful Theorem 3. If G, is a dense subgroup of a T, topological group G, with the cofinality

character X (G,) >= 1 , and if in G, the inverse of a CaucHY-sequence is again a CAUCHY- sequence, then the two completions are isomorphic. The same holds if we say ring instead of group. If we deal with topological skewfields, we ought to add the condition that the multiplicative inverse of a CAucHY-sequence not converging to zero is again a CAUCHY- sequence.

Proof. Observe that by theorem 1 the character of the dense subgroup G, is that of G,. Hence we may consider Caucwz-sequences of the same length X for the two completions. Since a CAucnY-sequence in G, is, by the definition of the relativized uniformity on G,, a CAUCHY-sequence in G,, t'he completion of G, is embedded in the completion of G,.

But since G, is dense in G,) every CAucHY-sequence in G, can be accompanied by a CAucHu-sequence in GI such that the difference of the two sequences converges to zero. This proves the isomorphism from GI onto G, . Clearly, the same proof can be estab- lished for rings and skewfields, for the embedding maintains the algebraic structures.

THE COFINAL CHARACTER OF UNIFORM SPAOES AND ORDERED FIELDS 381

8 3. Fields which are dense in their real closure By theorem 3, if k is an ordered field which is dense in its real closure, the com-

pletion of k contains a real closure of k. Moreover, as HAUSCHILD proved, these fields are characterized by the property to have a real closed completion. I n what follows, a cut ( A , B) in an ordered set k is understood to be a partition of k: k = A u B with A < B . The cases A = 0, B = 0 are not excluded. Thea may require a special treatment which is left to the reader.

Theorem 4. If k is an ordered field and t is tralzscendental over k , the following statements are equivalent :

(i) k is a real closed field; (ii) every extension of the given ordering of k to an ordering of k ( t ) is determined by

the induced ordering on th set k u { t } . Proof. (i) (ii): this is lemma 13.12 in [6], p. 182. It is due to the fact that an

(ii) + (i): if K is the real closure of k and k $. K , there is an a E K , a B k . Define

A = { x I z E K , x 5 a } , B = K - A , A’ = { x I z E K , x < a>, B’ = K - A’. These two cuts define two different orderings of K ( t ) (see proposition 5), which differ even on k( t ) (proposition 6). This yields a contradiction, since obviously A n k = = A ’ n k, B n k = B’n k.

To fill the gap in the proof, we give part of a solution of a problem of BOURBAKI [l], ex. 15, p. 46.

Propos i t ion 5. Let K be a real closed field; to every cut ( A , B) in K there exists one and only one ordering of K ( t ) , t transcendental, such that A < t < B .

Proof. The unicity is granted by the quoted lemma 13.12 of [6]. For the construc- tion of the ordering we distinguish two cases. 1) ( A , B ) is a gap in K . So for every f ( t ) E K [ t ] , f ( t ) $. 0 , we can find M f E A such that either f ( z ) > 0 for all z E A , z > M f , or f ( z ) < 0 for all z E A , z > M f . We define an ordering of K[ t ] with respect to the cut ( A , B) by

irreducible polynomial over a real closed field has a t most degree2.

two cuts, ( A , B) and (A’, I?’), in K as follows:

f ( t ) > O o f ( z ) > 0 for all zE A , z > M f . Verifications are trivial and we have A < t < B . This ordering can be extended to an ordering of K ( t ) and we remark without proof that in this case K ( t ) is archi- medean over K . 2) If ( A , B) is not a gap and B has a first element B, then A has no last element and the above construction works word by word. If A has a last element 01, then B has no first element and the modifications are obvious. I n both cases K ( t ) is not archimedean over K , the infinitesimals in K [ t ] being precisely the multiples of the irreducible polynomial of 01 or B, respectively. (If K c Kl are ordered fields, an element f E Kl is called infinitesimal with respect to K if I f 1 is smaller than any positive element of K ) .

382 PAUL HAFNER AND GUERINO YAZZOLA

Propos i t i on 0. If K i s the real closure of the ordered field k and if t is trarwcendental over k, then every ordering of k ( t ) extending the given ordering of k i s iaduced by pre- cisely one ordering of K ( t ) .

P r o of. Since the real closure of k ( t ) contains a real closure of k, every ordering of k ( t ) can be extended to an ordering of K ( t ) . It remains to be shown that different orderings of K ( t ) induce different orderings of k ( t ) . Let two different orderings of K ( t ) be defined by the cuts ( A , B) , (A’, B’) .with A c A’. Then C = A’n B + 0. By the construction in the proof of proposition 5, it will now be sufficient to exhibit a polynomial fo( t ) E k[t] which has precisely one root in C; fo( t ) will then be positive in the ordering of K ( t ) and negative with respect to the other. To this end, let F denote the set of all polynomials f ( t ) E k [ t ] of positive degree which have a root. in C . Clearly, F 0, so let f o E F have minimal degree. f , has only one root in C (otherwise, the derivative f h , which is of lower degree, would also belong to F) . The same reasoning shows that f , is monotone in C; so the last part of the previous proof implies

P ropos i t i on 7. Let K be the real cbsure of the ordered field k. To every (open or closed or semi-closed) interval I c K there exists a n irreducible polynomial f r E k[x], which i s monotone in I and which has precisely one (simple) root in I .

Theorem 8. For an ordered field k the following conditions are equivalent ( t trans- cendental over k, K the real closure of k) :

(i) k i s &nse in its real closure; (ii) if f ( t ) E k[ t ] i s such that there exist a , b E k with ! (a) f (b) < 0 , t h m to every

positive d f fc there exists c E k between a and b and such that 1 f ( c ) I < d ;

(iii) the completion of k i s real closed; (iv) the number of different extensions of the orderiszg of k to a n ordering of k ( t ) which

induce the same ordering on k u {t] i s at most 2 .

Proof . (i) e (iii) is Satz 14 in [7]. (i) - (ii): by contraposition. Suppose there are a , b , d E k and f ( t ) E k[ t ] with

f ( a ) f ( b ) < 0 and I f (c)I > d for all c E k between a and b . f has a root z E K of odd multiplicity between a and b and K contains an open interval I c [a , b) , containing x and such that f is monotone in I and I f (u) 1 < d for all t~ E I. So it is obvious that k cannot be dense in K .

(ii) => (i): also by contraposition. If k is not dense in K , there is an interval I, = (a , b) c K with a $: b , I , n k = 0 and we shall assume that I, is maximal (use ZORN or any other argument). Let now f r o = f be as in proposition 7. We denote by

XO the root of f in I,; xl, x2 elements of I , with xl < x, < x,; z1 resp. z, the greatest reap. smallest root of the derivative f’ to the left resp. to

the right of x,, if such a root exists; otherwise, let it be any element to the left resp. to the right of I,,.

THY: COFINAL CHARACTER OB UiiLFORM SPACES AND ORDERED FIELDS 383

Now, by the maximality of I,, , there are yl , y, E k such that z1 5 yl < x,, < y, 5 zg andfismonotonein (yl,y2].Sof(yl)f(yz) < OandJf(c) l > dfora l lcE (y l ,yz )nk and all positive d < min{ I f (xl) I , I f (x2) I}. But this is the negation of (ii) as it should be. (To clear up the above chaos of definitions, it may be helpful to draw a picture.)

(i) + (iv): k is dense in K . Let the cuts corresponding to two different orderings of K ( t ) be (A , B) and (A’, B’) where A‘ 2 A . Suppose now that these two orderings define the same ordering of k u {t}; by density, A‘ n B = {a} , a E K , a B k. So A’ has a as last element, B has a as first element and ( A , B) , (A’, B ) are the only cuts in K with this property.

(iv) * (i): This is the same as what has been done thus far and will be omitted.

Remarks . The equivalence of (i), (ii) and (iii) is due to HAIJSURILD [7], Satz 13 and Satz 14. The above proof, however, differs considerably from his. The idea of (iv) is to assign a number a(s) to every cu t s = ( A , B) of k: the number of different extensions of the given ordering of k to an ordering of k ( t ) such that A < t < B . We point out that n (s) = 1 , 2 or 00 in any case. If n (a) = 1 for all s , then k is real closed; if n(8) = 2 then both extensions corresponding t o s are not archimedean over k. Proposition7 also provides a proof of HAUSCHILD’S theorem 15.

$4. A field with cofinal character co,, It should be remarked that for an ordered field k the cofinal character of k as

uniform space and the cofinal character of k as ordered set coincide. Starting from an arbitrary ordered field k and any regular initial ordinal wp it is possible to con- struct an ordered field R with cofinal character w,, containing k (see [7], p. 65). For, let T = {ty},.<,,p be a set of independent transcendentals over k anddenote by T, (L w,,) the subset {tY}v<r. Then K = k(T) can be ordered in such a way that k(T,) < t, for all 1c 2 L . If k = Q, we denote the resulting ordered field by K,, - K,, has cardinality N,,.

Theorem9. Let k be any ordered field of cofinal character o,,, p > 0 . There is an order preserving isomorphism p : K,, + k such that p(K,) is cofinal in k.

Proof. Since p > 0, k is uncountable; therefore k has a dense transcendence base T’ (lemma 13.11 in [S]). By choosing a suitable subset T of T’ the theorem can be proved. SII~ORSICI [12] exhibits also a “least field with character w,,”.

The field K,, (or its additive group rather) may be used for a metrization of uniform spaces X with cofinal character w,,, p > 0 . The construction is as follows: let {Nl} l<wp be a nested base of the uniformspace X consisting of symmetric entourages. For x , yE X, x + y y define

e(x, y) = min{e I (2, y) B N ; for all n E N}. Now the distance d(x, y) may be defined by

d(x, x) = 0, d(x, Y) = ( t & , J - l , for 2 * y.

384 PAUL W N E R AND GUERINO MAZZOLA

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