Disjoint Unions of Topological Spaces and Choice

Math. Log. Quart. 44 (1998) 493 - 508

Disjoint Unions of Topological Spaces and Choice

Paul Howarda, Kyriakos Keremedisb, Herman Rubin", and Jean E. Rubind

a Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, U. S. A.l) Department of Mathematics, University of Aegean, Karlovasi, Samos 83200, Greece2) Department of Statistics, Purdue University, West Lafayette, IN 47907, U. S. A.3) Department of Mathematics, Purdue University, West Lafayette, IN 47907, U . S. A.4)

Abstract. in the absence of some form of the Axiom of Choice.

Mathematics Subject Classification: 03E25, 04A25, 54D10, 54D15.

Keywords: Axiom of Choice, Disjoint union of topological spaces, Compactness, Normal space, Hausdorff space.

We find properties of topological spaces which are not shared by disjoint unions

0 Introduction and terminology

This is a continuation of the study of the roll the Axiom of Choice plays in general topology. See also [l], [2], [3], and [4]. Our primary concern will be the use of the axiom of choice in proving properties of disjoint unions of topological spaces (see Definition 1, part 11). For example, in set theory with choice the disjoint union of metrizable topological spaces is a metrizable topological space. The usual proof of this fact begins with the choice of metrics for the component spaces. We will show that the use of some form of choice cannot be avoided in this proof and in fact without choice the disjoint union of metrizable spaces may not even be metacompact.

In Section 1 we show that many assertions about disjoint unions of topological spaces are equivalent to the axiom of multiple choice. Models of set theory and corresponding independence results are described in Section 2. In Section 3, we study the roll the Axiomof Choice plays in the properties of disjoint unions of collectionwise Hausdorff and collectionwise normal spaces.

We begin with the definitions of the symbols and terms we will be using

')e-mail: [email protected] ')e-mail: [email protected] 3, e-mail: hrubin@stat .purdue.edu *)e-mail: [email protected]

494 Paul Howard, Kyriakos Keremedis, Herman Rubin, and Jean E. Rubin

D e f i n i t i o n 1. Let (X, T ) be a topological space.

1. A family Ic of subsets of X is locally finite (1.f.) iff each point of X has a neighborhood meeting a finite number of elements of K.

2. X is paracompact iff X is Tz and every open cover U of X has a locally finite open refinement (1. f. 0. r . ) V . That is, V is a locally finite open cover of X and every member of V is included in a member of U .

3. A family Ic of subsets of X is point finite (p. f.) iff each element of X belongs to

4. X is metacompact iff each open cover U of X has an open point finite refinement

5. An open cover U = {U; : i E k } of X is shrinkable iff there exists an open cover : i E k } of non-empty sets such that vi C_ Ui for all i E k . V is also called a

only finitely many members of Ic.

(0. p.f. r.).

V = { shrinking of U.

6. X is a PFCS space iff every p.f. open cover of X is shrinkable.

7. D C_ X is discrete if every x E X has a neighborhood U such that IU n DI = 1. A set C P(X) is discrete if for all x E X, there is a neighborhood U of x such that U n A # 0 for at most one element A E C. (Thus, D C X is discrete iff { i d } : d E D } is discrete.)

8. X is collectionwise Hausdorff (cwH) iff if X is T2 and for every closed and discrete subset D of X , there exists a family U = {ud : d E D } of disjoint open sets such that d E ud for all d E D.

9. X is collectionwise Hausdorff with respect t o the base B (cwH(B)) in case U C B , where U is defined as in 8.

10. X is collectionwise normal (cwN) iff X is T2 and for every discrete C of pairwise disjoint closed subsets of X, there exists a family U = {UA : A E C} of disjoint open sets such that A C UA for all A E C. (cwN(B) is defined similarly to cwH(B) in 9.)

11. The disjoint union of the pairwise disjoint family {(Xi , T i ) : i E k } of topological spaces is the space X = U { X i : i E k} such that 0 is open in X iff 0 n Xi is open in Xi for all i E k .

It is clear from Definition 1 that

(1)

None of the implications in (1) are reversible. It is provable in ZFo , (ZF minus Founda- tion) that paracompact spaces are normal. (For example, the proof of Theorem 20.10 givenin [12, p. 1471 goes through in ZFo with some minor changes.) However, this conclusion does not hold for detacompact spaces. Dieudonnk's Plank (Example 89 in [ll, p. 1081 is an example of a metacompact, non-normal space. Any infinite set X endowed with the discrete topology is an example of a non-compact, paracompact space.

Below we give our notation for the principles we use and their precise statements:

T2 + compact j paracompact j metacompact.

Disjoint Unions of Topological Spaces and Choice 495

D e f i n i t i o n 2.

1. The Axiom of Choice (AC). For every family A = {Ai : i E k} of non-empty pairwise disjoint sets there exists a set C which consists of one and only one element from each element of A.

2. The Multiple Choice Ax iom (MC). For every family A = {A; : i 6 k} of non- empty pairwise disjoint sets there exists a family F = {Fi : i E k} of finite non-empty sets such that for every i E k, Fi 5 Ai.

3. The Countable Multiple Choice Axiom (CMC). MC restricted to a countable family of sets.

4. w-MC. For every family A = {Ai : i E k} of non-empty pairwise disjoint sets there exists a family F = {Fi : i E k} of countable non-empty sets such that for every i E k, Fi Ai.

5. CMC,. CMC restricted to countable sets (form 350 in [5]). 6. MP. Every metric space is paracompact.

7 . MM. Every metric space is metacompact.

8 . DUM. The disjoint union of metrizable spaces is metrizable.

9 . DUP. The disjoint union of paracompact spaces is paracompact.

10. DUMET. The disjoint union of metacompact spaces is metacompact.

11. DUPFCS. The disjoint union of PFCS spaces is PFCS.

12. DUPN. The disjoint union of paracompact spaces is normal.

13. DUMN. The disjoint union of metrizable spaces is normal.

14. vun Douwen’s Choice Principle (vDCP(w)). If A = {Ai : i E w } is a family of non-empty disjoint sets and f a function such that for each i E w , f ( i ) is an ordering of Ai of type 2Z (= the integers), then A has a choice function.

15. DUN. The disjoint union of normal spaces is normal.

16. DUcwH. The disjoint union of cwH spaces is cwH.

17. DUcwN. The disjoint union of cwN spaces is cwN.

18. DUcwNN. The disjoint union of cwN spaces is normal.

All the proofs below are in ZF’.

1 Disjoint unions and multiple choice

In this section we show that many “disjoint union” theorems are equivalent t o MC. As a consequence of the following well known lemma, these theorems are equivalent to AC in ZF.

L e m m a 1 ([S]). I n ZF, AC is equivalent t o MC

It is also known (see [S]), that in ZF’, MC does not imply AC.


T h e o r e m 1. E a c h of t he fol lowing are equivalent t o MC:

(i) DUP. (ii) DUMET. (iii) DUPFCS. (iv) I j X is t h e dis joint u n i o n of a f a m i l y {Xi : i E k} of compact T2 spaces, t h e n

every open cover U of X has a n open re f inemen t U such tha t for every i E k on ly a f i n i t e n u m b e r of e l emen t s of U mee t Xi non-tr iv ial ly .

(v) E v e r y open covering of a topological space X which is t h e dis joint u n i o n o f spaces Xi, i E k, each of which i s t h e one point compactzfication of a discrete space, can be expressed as a wel l ordered u n i o n of se t s U , 'P(X), where each U , is locally f i n i t e ( p o i n t f i n i t e ) .

(vi) E v e r y open covering of a topological space can be expressed as t h e wel l ordered u n i o n of locally f i n i t e ( p o i n t f i n i t e ) sets .

If w e restr ic t (i) through (v) t o countable f a m i l i e s of topological spaces, t h e n t h e result ing s t a t e m e n t s are equivalent t o CMC.

P r o o f .

MC=+(i). Fix a family X = {Xi : i E k} of pairwise disjoint paracompact spaces and let X be their disjoint union. Fix an open cover U of X . W. 1.0. g. we assume that each member of U meets just one member of K . Put Ui = {u E U u n Xi # S} and Ai = {V : V is a l .f .o. r . of Ui}. As Xi is paracompact, Ai # 8. Use MC to obtain a family F = {Fi : i E k} of finite sets satisfying Fi C Ai, i E k. Then, V = u { u F i : i E k } i s a l . f . o . r . o f U .

(i)+MC. Fix a family A = {A; : i E k } of pairwise disjoint non-empty sets. W . l . o . g . we assume that each Ai is infinite. Let {yi : i E k } be distinct sets so that for each i E k, yi 6 U A . Put the discrete topology on Ad and let A: denote the one point compactification of Ai by adjoining the point yi to Ai. Let X be the disjoint union of the family {A: : i E k}. As Af is paracompact (Af is a compact Tz space) our hypothesis implies that X is of the same kind. Let V be a l . f .o . r . of the open cover U = { U : U is a neighborhood of y; in A t , U # Af for some i E k } . Since U is a l .f . open cover, it follows that Fi = U{Ai\v : v E V A yi E v} is finite and non-empty. Thus, F = {Fi : i E k} satisfies MC for A.

MCe(i i ) . This can be proved exactly as in MCe( i ) . MC+(iii) and MC+(iv) can be proved as in MC+(i).

(iii)+MC. Fix a family A = {Ai : i E k} of infinite pairwise disjoint sets and let Af and X be as in the proof of (i)=+MC. Clearly, Af is a PFCS space. Thus, by (iii), X is also a PFCS space. Let U be a shrinking of the p . f . open cover U = {u : u = Ai or u = A:, i E k } of X. Clearly Fi = v, v E U and c Ai is a finite non-empty subset of Ai. Hence, F = {Fi : i E k} satisfies MC for A .

(iv)+MC. Let A and X be as in (iii)+MC. Let U be a refinement of the open cover U = {AT\{a} : a E A i , i E k } of X which is guaranteed by (iv). Clearly, Fi = U{A:\v : v E UAyi E v} is a non-empty finite subset of Ai and F = {Fi : i E k} satisfies MC for A.


MC+(vi). Since MC is equivalent to the statement “Every set is the union of a well ordered family of finite sets” (see [7]), it is clear that MC implies (vi). It is also clear that (vi) implies (v).

(v)+MC. Let A = {A* : i E k} be a family of disjoint non-empty infinite sets. Let X be as in (i)+MC and let G be the set of proper neighborhoods of y,, for some i E k. (If g E G, then there is an i E k such that g is a neighborhood of y, and g 5 Af .) G is an open covering of X , so by (v), G = u{ U, : Q E T}, where each U, is locally (point) finite. For i E k, let Q, be the smallest rw such that y, E UU,. (Notice that y, E g, for g E G iff every neighborhood of y, has a non-empty intersection with g , so it does not matter whether the U,’s are locally finite or point finite.) It follows that there are only a finite number of sets, gl,gz,.. . ,gn, in U,, such that yi E gj, for j = 1 , 2 , . . . , n . Let h, = ny=, gj. Then hi is a neighborhood of y; and hi 5 A t . Consequently, A, \ hi is a finite non-empty subset of Ai so that it follows

0

R e m a r k 1. We remark here that the space X in (i)*MC is normal (also com- pletely normal, i .e. every subspace of it is normal), but if MC does not hold for the family A , then X is neither paracompact nor metacompact. Also, since At in the proof of (i)=+MC is a compact TZ space, it follows that MC is equivalent t o the statement:

that { A , \ hi : i E k} is a multiple choice set for A.

(1) The disjoint union of compact T2 spaces is paracompact.

If in (1) we replace disjoint union with product and paracompact with compact, then the resulting statement, Tychonoff’s Compactness Theorem for compact Hausdorff spaces,

(2) The product of compact T2 spaces is compact,

is equivalent to the Boolean Prime Ideal Theorem (see [8]), and (2) clearly implies:

( 3 ) The product of compact T2 spaces is paracompact

We do not know if (3) implies (2). Similarly, the statements

(4) The disjoint union of compact Tz spaces is metacompact

and

(5) The disjoint union of compact T2 spaces is a PFCS space

are equivalent to MC, and (3) implies

(6) The product of compact Tz spaces is metacompact.

We do not know the relationship between ( 3 ) , (6) and

(7) The product of compact Tz spaces is PFCS.

(However, the product of paracompact spaces may not be normal, and therefore, may not be paracompact (see [lo]).

R e m a r k 2. Working as in Theorem 2.2 from [l] one can easily establish that DUPN implies vDCP(w). Hence DUPN cannot be proved in ZF without AC. Since paracompact spaces are normal (without appealing to AC) it follows that DUP implies


DUPN and consequently, MC implies DUPN. In the next theorem using ideas from [4] we show that the converse is also true. In fact we could deduce it as corollary to Theorem 1 of [4 ] , but we won't do it. We prefer to give a new proof and exploit some new ideas.

T h e o r e m 2. DUPN can be added to the list of Theorem 1. P r o o f . In view of Remark 2, it suffices to show that DUPN implies MC. Let

A = { A j : i E k} be a family of infinite pairwise disjoint sets. Let A = {ui : i E k} and B = { b , : i E k} be any two disjoint sets, each of which is disjoint from Ud. Let Ti be the topology on X , = {a$ , b i } U [A;]'" (where [Ail<" is the set of all finite subsets of A i ) , which is generated by

Bi = {C,(ai) : y E [A;]'"} U {D,(bi) : w E [A;]'"} U {{z} : z E [Ail'"},

where

C Y ( U i ) = {Ui} u {z E [Ail<" : 2 2 y}

and

D,(b,) = { b j } U {Z E [Ail'" : c n w = 8). C l a i m 1. Xi is paracompact. P r o o f of C l a i m 1. First we show that Xi is a Tz space. Fix z , y E X i . We

consider the following cases. C a s e 1. z, y E { ai, b i } . Assume that z = a, and y = b; . Then for any w E [Ail'",

C w ( a , ) and Dw(bi) are disjoint neighborhoods of z and y, respectively. C a s e 2. Assume z = a, and y E [Ail<". Let w E [Ail<" be disjoint from y.

Then {y} and Cw(u, ) are the required disjoint neighborhoods. If z = b , , then {y} and D, ( b i ) are the required disjoint neighborhoods.

Then {z},{y} are disjoint neighborhoods of z and y, respectively.

C a s e 3. z , y E

Thus, Xi is a Tz space as required. To see that ( X i , Z ) is paracompact, we fix an open cover U of X i . Assume

E(ai ) , O(bi) E U are neighborhoods of ai and bi, respectively. Then it follows easily that V = { E ( a i ) , O(b;) } U ({{z} : 3: E [A*]<"} is a locally finite open refinement of u. 0

C l a i m 2. If Cy(a i ) and D,(bi) are disjoint neighborhoods of ai and bi , then

P r o o f of C l a i m 2. If y r l 20 = 8, then y is in C, (a i ) n D,(bi) which is a contradiction. 0

Let X be the disjoint union-of the family { X i : i E k } . Then X , in view of the hypothesis, is a normal space and A and B are closed and disjoint sets in X . Hence, there exists disjoint open sets OA _> A and QB _> B. For every i E k put

W ~ U # O .

E ~ , ( a i ) = OA n X i , OB,(bi) = QB n X i , 2i = {Y E [A]'" Cy(ai) E ~ , ( a i ) } , Wi = {W E [Ail'" : D,(bi) 5 O g , ( b i ) } ,

ni = min{ly( : y E 2i}, Zi,,, = {y E 2i : Iyl = ni}.


C l a i m 3. Either Z,,,, i s finite, or

(*) P r o o f of C l a i m 3. Assume, to the contrary, that Zi,,, is infinite and (*) fails.

We shall construct inductively a set ( 2 0 0 , w1,. . . , w,,+l} of pairwise disjoint finite non-empty sets wj 5 UWi such that the set

there exists a finite Q C_ UWi such that f o r all w E Wi, Q n w # 0.

P = {y E Zj>,, : wj c y for all j 5 ni + 1)

is infinite. Then any y E P will satisfy ni = (y( 2 I u{wv : contradiction proves the claim.

from Claim 2, that each y E Z,,,, meets non-trivially wo. Hence,

4 ni + 1}1 > ni. This

Fix wo E Wi. Since D,o(bi) is disjoint from every C,(ai), y E 2i,0, , it follows

Z i , n , = U{{y E Zi,,, : 20' n y = W } : w c W O A w # 8) and there is a non-empty w E wo such that H I = {y E Zi,,, : w c y} is infinite. Put wo = w. Assume that pairwise disjoint and non-empty sets W O , w1, . . . , wn have been chosen so that the set {y E Zi,,, : W O , 2 0 1 , . . . , w, c y} is infinite. By the negation of (*), there exists w" E Wi such that (wOUwlU...Uw,)nw" = 0. By Claim 2 again, there exists a non-empty w w n such that Hn = {y E Zi,n, : WO, ~ 1 , . . . , w,, w C y} is infinite. This terminates the induction and the proof of the claim. 0

C 1 a i m 4. Assume 2i,nc is infinite and let 10 be the least integer for which there is a Q E [UWiI'o satisfying (*). Then IQi,loI < w , where

Put w,+1 = 20.

Qi, lo = { Q E [UWi]1° : Q satisfies (*)}. P r o o f o f C 1 a i m 4. Assume, to the contrary, that Qi,lo is infinite. We construct

inductively a finite sequence of finite pairwise disjoint sets 200, w1,. . . , w,, for some T E w , with the property that W = Ul=o wi has cardinality lo and infinitely many elements of Qi , lo include W . This contradiction will establish the claim.

Fix wo E Wi. As wo and each member of Qi, lo meets wo non-trivially, it follows that there exists a non-empty wo c wo such that KO = {Q E Qi,lo : wo & Q} is infinite. If lwol = l o , then P = 0 and the induction terminates.

Assume that pairwise disjoint finite non-empty sets W O , w1, . . . , w, have been chosen so that JwoUwl U...U w,[ < 10 and K, = {Q E Kn-l : wO,wl,. ..,w, c Q} is infinite. Fix wn+' E W, such that (woUwlU Uw,)nwn+l = 0. (Such a wn+l E Wi exists, for otherwise, wo U w1 U . . . U w, will satisfy (*) with (WO U w1 U . . .U w,( < l o . ) By the hypothesis again, there exists w,+1 C UP+' such that the set

Kn+l = { Q E K n : ~ 0 , ~ 1 , . . . , wn, wn+1 C - Q } is infinite, terminating the induction and the proof of the claim.

By Claims 3 and 4, it follows that F = {F, : i E k} where,

UZi,,, if Z,,,, is finite,

satisfies MC for d, finishing the proof of the theorem

0

0

Since the space Xi of Theorem 2 is also normal, cwH, and cwN, we get as a corollary to Theorem 2:


C o r o l l a r y ([4]). T h e foi lowing can be added t o the list of T h e o r e m 1: (i) DUN.

(ii) DUcwH. (iii) DUcwN. (iv) DUcwNN.

It is easy to see that closed subspaces of paracompact (metacompact) are also paracompact (metacompact) spaces without appealing to AC. The next result shows that CMC is needed for F,, sets (countable unions of closed subsets).

T h e o r e m 3. T h e fol lowing are equivalent:

(i) CMC (ii) F,, subsets of paracompact spaces are paracompact. (iii) F,, subsets of me tacompac t spaces are me tacompac t .

P r o o f . (i)+(ii). Follow the proof of Theorem 20.12 in [12, p. 1481. (ii)+(i). Fix a family A = {A i : i E w} of disjoint non-empty sets. Note that if

X" is the one point compactification of X as given in (i)+MC of Theorem 1, then X is F,, in X' and thus, paracompact. Continue the proof as in Theorem 1.

(i)+(iii). Let (X, 7') be metacornpact, let G = u{Gn : n E w} be an F,, set in X , and let U be an open cover of G. For every u E U pick v, E T such that u = u, n G. (Note that we do not need AC in order to pick the element v,. We can pick all such elements and then take their union. Clearly the union is such an element.) For every R E w put U, = {vu : u E U A v, n Gn # 0) U {X\G,}. Clearly Un is an open cover of X. As X is metacompact A, = {W : W is an 0.p.f . r . of U , } # 0. By CMC, there is a family F = { F , C A, : n E w} of finite non-empty sets. For each n E w , let B, = {G n s \ Urn<, G, : s E U F, A s n G, # 0). Then B, is p. f. and each element of B, is open in G. We claim that B = UnEw B, is an open p.f . refinement of U that covers G . To show that B covers G , suppose 2 E G. Take R to be the smallest natural number such that I E G,. It follows from the construction of F, that there is an s E U F, such that I E s. Consequently, z E US.

0

DUM implies DUMN because metric spaces are normal in ZFO. We do not know if DUMN implies DUM. However, in Theorem 4 we show that DUM + w-MC iff MC and in Theorem 5 we show that DUMN + w-MC iff MC.

The proof that (iii) implies (i) is similar to the proof that (ii) implies (i).

T h e o r e m 4 . DUM + w-MC ZflMC. P r o o f .

(e). That MC implies w-MC is obvious. To see that MC implies DUM, fix X = {Xi : i E k ) a disjoint family of metrizable spaces. For every i E k let Ai denote the set of all metrics on Xi producing its topology. Put A = { A , : i E k ) and use MC to find for each i E k a finite subset Gi = {ml, m2,. . . , m,,} of Ai . For all I, y in X i , let d,(z, y) be defined as follows:


Then d, produces the topology of X i . Clearly, the function d : U X x U X - R with

1 if t E X i and y @ X i for some i E k, d(z ’ ’) = { d; (z , y) otherwise,

is a metric on U X producing the topology of the disjoint union on X = U X . Thus, X is metrizable as required.

By w-MC we may assume that the members of A are countably infinite. Let X be as in Theorem 1, (i)*MC. Since each A: is metrizable, it follows by DUM that X is also metrizable. Let d be a metric on X producing its topology. For every i E k let ni = min{n : A; # D(yi, l / n ) n A * } , where D(yi , l / n ) is the open ball of radius 1/n centered at yi. Clearly, Fi = Ai\D(yi, l/ni) is a finite non-empty subset of Ai and

O

(+). Fix a family A = {Ai : i E k} of pairwise disjoint non-empty sets.

F = {Fi : i E k} satisfies MC for A. C o r o l l a r y . DUM implies CMC,. P r o o f . The proof is similar to the second part of the proof of Theorem 4. T h e o r e m 5. DUMN + w-MC iff MC. P r o o f . (e). It is shown in Theorem 4 that MC implies DUM. The result follows because

DUM implies DUMN and MC implies w-MC. (+). Let A = {Ai : i E k} be a family of pairwise disjoint non-empty sets.

By w-MC we may assume that the members of A are countably infinite. If Ai is countable so is [AilcW. Let (Xi, T,) be the corresponding topological space as defined in Theorem 2. The topology T, as defined in Theorem 2, clearly has a countable base. It is shown in Theorem 2 that Xi is paracompact, and, therefore, regular. Thus, it follows from [2, Corollary 4.81 that Xi is metrizable. Using DUMN it follows that the disjoint union of the Xi’s is normal. Using the same ideas as in Theorem 2, we can

0

R e m a r k 3. In the Cohen-Pincus model M 1( (q)) (see [5] and [9]) w-MC holds. Since MC is false in this model, it follows from Theorem 4 that DUM is also false. Thus, w-MC does not imply DUM.

0

construct a multiple choice function for A.

2 Examples and models

E x a m p 1 e 1 (A non-metrizable, non-paracompact countable disjoint union of paracompact spaces). The space L of Theorem 2.2 from [I] is clearly a non-normal (hence, non-paracompact) countable disjoint union of paracompact spaces.

E x a m p 1 e 2 (A metrizable non-paracompact countable disjoint union of paracompact spaces). Let N be the permutation model with set of atoms A = U{Q, : n E w } , where Qn = {a,,q : q E 0).

< am,p iff n < m, or n = m and q < p . The group of permutations G is the group of all permutations on A which are a rational translation on Q,, i .e. if cp E G, then c p l Q , ( ~ ~ , ~ ) = u , , ~ + ~ , for some r , E Q, and supports are finite. Let d be the metric on A given by

Suppose < is the lexicographic ordering on A , i. e.


GOOD, TREE and WATSON [3] prove that (A , d ) is a linearly ordered, zero dimensional, metric space, but is not paracompact (MP is false). Each Q,, being a second countable metric space, is paracompact, and the disjoint union topology on A coincides with the metric topology which is normal. Hence, A can be considered as a normal non- paracompact disjoint union of paracompact spaces.

The disjoint union of the Q,’S is also not metacompact. The set of intervals

V = { ( ~ , b ) : ( 3 n E w ) ( ~ , b E Q n A a < b ) }

is an open cover without a point finite refinement in the model. (Suppose U is a point finite refinement of V with support E and Q, n E = 0. Then for some U E U , U 5 ( a , b ) Qn. Choose 2 f Qn and an interval1 ( c , d ) C ( a , b ) such that x E ( c , d ) c U . Then it is easy to see that x is contained in a countably infinite number of translates of V . ) Thus, MM is false in N .

DUMN is false in N . For each n E w , let X, = Qn U {cn , d,}, where cn and d, are distinct sets in the model which are not in A, and let <, be the ordering on Qn extended so that c , is the smallest element and d, is the largest element. (Xn, <,) is metrizable because it is the subspace of a metric space. (It is isomorphic to the closed interval of rational numbers [0, 11.) Let X be the disjoint union of the Xnls. Let C = {c, : n E w } and D = {d, : n f w } . The sets C and D are disjoint closed sets in X. If X were normal there would be disjoint open sets U and V containing C and D, respectively, such that their closures are disjoint. For each n E w , let Y, = Qn \ u. Then Y, 5 Q,. Since supports are finite, it is easy to show that there is no such function f : n - Y, in the model. Thus, DUMN is false, and since DUM implies DUMN, DUM is also false.

The Axiom of Choice for pairs, C(oo,2), is also false in N. First note that if a permutation leaves any element of any Q, fixed, then it has to leave the whole set fixed, because the permutations are translations. Therefore, every subset of each Qn is in the model. Let B be the set of all unordered pairs { C , D } , where C c Qn and D C Q n , C # D , for some n E w. The set B has empty support so it is in the model. Suppose f is a choice function on B with support E. Since E is finite, there is an n E w such that Eng, = 0. Let 0, = {un,2i+l : i f iz} and let En = {a,,zi : i E Z}. Then {On, En} E B. Let x be a permutation in fix(E) such that x(am,,) = if m # n, and x(a,,,) = a,,,.+l. Then x leaves f and {On, En} fixed, but interchanges 0, and En, which is a contradiction. Thus, C(oo, 2) is false in the model. Since, the Boolean Prime Ideal Theorem and the Ordering Principle (every set can be linearly ordered) each imply C(w, 2), they are also false.

However, in N, every infinite set has a countably infinite subset. Let X be an infinite non-well-orderable set with support E and let x1 E X be such that E is not a support of XI. (Such an element exists because otherwise, X can be well ordered.) Let El be a support of 2 1 . For n, m f w and p , q E Q, we define x;t E G as follows:

and

For am,p E A ,

if m # n , if m = n.

We claim that there is an n E w and q f Q such that xy f fixE and x ; t ( x l ) # 21. (Otherwise, E would be a support of $1.) We shall show that S = {x;(x1) : q E Q}


is a countably infinite subset of X in N . S has support El SO it is in N . Also, S C X , as BY E fixE. It remains to show that S is infinite. Let H = { q E Q : xg(z1) = 2 1 ) .

It follows from the definition of n that H is a proper subgroup of Q. We shall show that S is infinite by showing that the factor group Q / H is infinite. Suppose the factor group has order n and suppose p $ H . Then n(p/n) + H = H , because n is the order of Q / H , but n(p /n ) + H = p + H , which is a contradiction because p @ H .

Thus, we have shown that

N I= “Every infinite set has a countably infinite subset” +7MM + 7DUMN + -6’(00,2).

E x a m p 1 e 3. In the Mostowski linearly ordered model, the disjoint union of metrizable spaces is metrizable.

P r o o f . Let ( A , <) be the set of atoms in the the Mostowski linearly ordered model M . ( A is countable and < is a dense linear ordering on A without first or last elements.) For notational convenience we add first and last elements -ca and 00 to the ordering ( A , <). Let G denote the group of order automorphisms of ( A , <), and for each E A let GE = {‘p E G : ‘p fixes E pointwise}. Supports in this model are finite subsets of A . For each 2 E M we denote the minimal support of 2 by sup(x). The crucial lemma is

L e m m a 2. If (X, T) i s a metr izable topological space in M , t h e n there i s a m e t r i c m o n X such tha t

(a) m is in M , (b) T i s t h e topology induced by m, (c) sup(m) = sup(X, T).

We shall show first that it follows from Lemma 2 that DUM is true in M . Assume that {(Xi, I;.) : i E I<} is a collection of disjoint metrizable topological spaces in M with support D. For each i E I< choose a metric mi on Xi so that mi is in M , mi induces the topology Ti and sup(mi) = sup(Xi , Ti). (We may assume w. 1.0. g. that each mi is bounded by one.) It follows from the last of these properties that for all cp f Go, cp(mi) = mi if and only if ‘p(Xi,Ti) = (Xi,Ti). (This would not be true, for example, in the basic Fraenkel model where “cp fixes sup(z) pointwise” implies “cp fixes d’, but not conversely.) Therefore, the function (Xi, Ti) H mi has support D so it is in M . We can use this function to define a metric m on UiEK Xi which induces the disjoint union topology (m(x, y) = rn;(x, y) if 2, y E Xi for some i E I<, and m(z, y) = 1 otherwise).

To prove Lemma 2, let (X,T) be a metrizable topological space in M and let d be a metric on X which is in M and which induces the topology T. If cp E G fixes d then cp fixes X and T and therefore sup(X,T) C sup(d). Assuming that sup(X,T) # sup(d), it suffices to construct a metric m satisfying (a) and (b) from the statement of Lemma 2 and

(c’) SUP(m) 5 sup(d). By our assumption that sup(X, T ) # sup(d) there is an element t E sup(d)\sup(X, T ) . Let e o , e l be the unique elements of sup(d) U {--00,00} such that eo < t < e l


and both of the open intervals ( e 0 , t ) and ( t , e l ) are disjoint from sup(d). Let E = sup(d) \ { t } . Since E C, sup(d), the proof can be completed by constructing a metric m with support C E and satisfying (a) and (b).

At this point we need a well known property of linear orders: L e m m a 3. I f (C, and (D,<Z) are countable dense l inear orders wi thout

f irs t o r last e l e m e n t s and C’ and D‘ are f i n i t e subsets of C and D , respectively, wi th IC’I = ID’I, t h e n there is a n order i s o m o r p h i s m f f r o m C on to D such tha t f ( C ’ ) = D’.

Using Lemma 3, let A’ be an order isomorphism from ( e 0 , e l ) onto ( e 0 , t ) and define A : A - A \ [t , e l ) by

A(u) = { t ’ ( U ) if a E ( e o , e 1 ) , otherwise.

We first note that for any x E M , there is a 11, E GE such that 11, agrees with A on sup(x). (This is a consequence of Lemma 3. ) Further, if 11, and cp are in GE and 11, and cp agree with A on sup(x), then $(I) = cp(z). This allows us to define a function A* from the topological space X to itself by A*(I) = $(x), where 11, E GE and $ agrees with A on sup(%). We leave to the reader the proof that A* is one to one and onto the set X’ = {x E X : sup(”) n [ t , e l ) = a}. There are two natural ways of putting a topology on the set X’:

TI = {U’ : U E T } , where U’ = {A*(z) : x E U )

and

r , = { v n X * : U E T } .

The topology TI is the topology under which the function A * : X - X’ is a homeomorphism (between the spaces ( X , T ) and ( X , T l ) ) . The topology Tz is the relative topology on X * inherited from ( X , T ) and is therefore the topology induced by the metric d restricted to X’. We will call this restricted metric d’.

We are now able to outline our plan for finding a metric m for the topology T with support contained in E : We will show that TI = Tz. It follows that if we “pull the metric d’ back to X using (A*)-”’ we get a metric which induces the topology T on X . (This “pull back” of d” is the metric m defined below.) The metric d* has support E U { t } , however in pulling d’ back to m, t is identified with e l and therefore m will have support E . The most difficult part of the argument is the proof that TI = T2. We postpone this part and begin with the definition of m:

m(z , Y) = d(A*(I) , A*(Y)>.

The fact that m is a metric follows from the fact that d is a metric. To prove the triangle inequality, for example, choose x , y , t E X and 11, E GE such that 11, agrees with A on sup(x) U sup(y) U sup(z). Then

m(zc, Y) + m(Y, 2) = d(11,(c), $(Y)) + 4 1 1 , ( Y ) , 11,(z)) 2 411,(x), 11,(z)) = 4 x 7 2).

We now argue that for all r ] E G E , r ] fixes m. This can be accomplished by showing that for all I, y E X , m(q(x), r](y)) = m(z , y). Assume r ] E GE and c, y E X . Choose


'p E GE such that cp agrees with A on sup(") U sup(y) U sup(v(z)) U sup(v(y)). By

The permutation ' p ~ c p - ~ takes cp(z) to cp(v(z)) and cp(y) to cp(q(y)). Further, since the range of A is A \ [t, e l ) , the four sets sup(cp(z)) = cp(sup(z)), sup(cp(y)) = cp(sup(y)),

A \ f t , e l ) . Therefore, we can find a permutation u E GE such that u agrees with cpvcp-' on sup(cp(z)) u sup(cp(y)) and such that u fixes [ t , e l ) pointwise. It follows that u(cp(z)) = cp(v(z)) and a(cp(y)) = cp(v(y)). Also, since u fixes E U { t } ,

the definition of m, m(z , Y) = d(cp(z), cp(Y)> and m(v(z ) , V(Y)) = d(cp(v(z)), P ( V ( Y ) ) ) .

sup(cp(v(z))) = cp(suP(v(z)))l and sup(cp(v(y))) = cp(SUP(V(Y)>) are all subsets of

44 = d. Therefore d(cp(z), cp(Y>) = + ( c p ( 4 ) , 4 c p ( Y ) ) ) = d(P(V(")), cp(V(Y))) . Hence m(z , Y) = m(v(z ) , rl(Y)).

We conclude that (a) and (c') are true of m. We shall complete the proof of Lemma 2 and also show that TI = T2 by proving

(i) Every U E T is in the topology induced by m. Assume U E T and

part (b) of Lemma 2 , T is the topology induced by m. We first show

E U . Choose 'p E GE so that cp agrees with A on the set sup(z)usup(U). Since cp fixes T , p(z) E cp(U) E T . Since the metric d induces T , there is an E > 0 such that the open ball B1 = {y E X : d(cp(z),y) < E } C cp(U). We claim that B2 = {y E X : m(z,y) < E } U . For suppose y E B2. Then (by definition of m) d(cp'(z), cp'(y)) < E , where cp' agrees with A on sup(") usup(y) Usup(U). Since cp'(z) = cp(z), cp'(y) E B1. Therefore, cp'(y) E cp(U) = cp'(U) and hence y E U .

Now we show (ii) If U is open in the topology induced by m, then U E T . It suffices to show that for every z E X and every E > 0,

(*) (36 > 0) ( 2 E x : d(z , 2 ) < 6) C ( 2 E x : m(2, z) < E } .

Assume that there is an z E X and an E > 0 for which (*) is false. Fix an s in the interval ( t , e l ) so that sup(z) n [s , e l ) = 8. The failure of (*) gives, for each positive n E w , an element 2, of X such that d ( z , z,) < ~ / n and m(z , 2,) > E . (The function n H 2, may not be in M . )

C 1 a i m . We may assume that sup(z,) n [s, e l ) = 0. If this is not the case, choose s' E ( t , s ) which also satisfies sup(") n [s', e l ) = 0 .

Let y' be an order isomorphism of (s ' , e l ) for which y'(sup(z,) n ( s ' , e l ) ) (s ' , s ) . The permutation y which agrees with y' on ( s ' , e l ) and is the identity outside of (s', e l ) then has the property that sup(y(z,)) n [s , e l ) = ~(sup(z , ) ) n [ s , e l ) = 0 . In addition, y fixes E U { t } U sup(z) pointwise. This means that y fixes d , m, and z. Hence d(z , y(zn)) < & / n and m(z , y(zn)) > E . We may therefore replace 2, by ~ ( 2 , ) .

This proves the claim. Choose cp E GE which agrees with A on ( e 0 , s ) . (This can be accomplished by

using Lemma 2 to obtain an order isomorphism from [s , e l ) onto [A(s), e l ) . Then define p(u) = p ( u ) for u E [s , e l ) and cp(a) = A(u) otherwise.) For n E w \ {0}, define 2, = {$(z,) : $ fixes sup(d) U sup(z) U { s } pointwise}. Then

(A) 2, has support sup(d) U sup(z) U {s}. (B) For all w E Z,, d(x ,w) < &/n. (Since w = +(z,), where $ fixes z and d and

d(", Z n ) < E/n.)


(C) For all w E Zn, m ( ~ , W ) > E .

(D) For all w E Z,, sup(w) n [s, e l ) = 8. By (D), for all w E Z,, p (which is in G E ) agrees with A on sup(w) Usup(z). Hence, using (C) and the definition of m we conclude that

(El d ( ' p ( 4 , 4 W ) ) > E .

By (A), uFZp=, 2, E M . By (B), x is in the T closure of uF=, 2, since d induces T . Similarly, by (E), p(z) is not in the T closure of ~ ( u , " , ~ Zn). This is a contradiction since 'p E GE and therefore fixes T . This completes the proof of (b) and the statement of Example 3. 0

3 Disjoint unions of collectionwise Hausdorff spaces, collectionwise normal spaces and the axiom of choice

T h e o r e m 6. The following are equivalent:

(i) AC. (ii) Zfa topological space ( X , T ) is cwH, then it i s cwH(B) f o r every basis B . (iii) The disjoint union ofcwH spaces is cwH(B) f o r any basis B. (iv) Zf a topological space ( X , T ) i s cwN, then it i s cwH(B) f o r a n y basis B .

P r o o f . (i)+(ii). Let ( X , T ) be cwH, G = {gi : i E k} a closed discrete subset of X and

B a basis for X. As X is cwH, there exists a disjoint family {Oi : i E k} C T such that gi E Oi for all i E k. Since B is a basis it follows that

Q i = { b E B : gi E b Oi} # 8. Let Q = {Qi : i E k} and v be a choice function on Q. Clearly V = { v ( i ) : i E k} C B is the required family of open sets.

The proofs (ii)+(iii) and (iii)+(iv) are straightforward. (iv)+(i). Fix a family A = { A i : i E w } of infinite pairwise disjoint sets. For each

A E A, let X A be a set not in A U U A and chosen so that the function A H X A is one-to-one. (To be specific, we could take X A to be the ordered pair ( A , A).) For each A E A, we define a topological space (YA, IA) as follows: YA = { X A } U (w x A ) and 7 A is the topology with basis sets

BA = { { ( n , U ) } : ( n , U ) E w X A } U { { X A } U Z : (w \m) x A Z E w x A for some rn E U } .

Let Y = UAEa YA and let 7 be the disjoint union topology on Y . We claim that Y is cwiy. Indeed, let Z be a discrete collection of closed sets. I t can be readily verified:

(1) For every A E A either X A $ z for all z E 2, or if X A E Z* for some Z* E 2, then there exists nz= E w such that for every z E Z \ { z * } , z 2 nz* x A .

For every z E 2, let

(2) U, = U { ( u x A ) n z : A E A A z A $ ~ } U ( U { Z U ( Y A \ U Z ) : A E A A x A E ~ } ) .


In view of (1) and (2)) it follows that 24 = {Uz : z E Z } is a disjoint family of open sets separating the members of 2. It is easy to see that each YA is Tz, so that Y is also Tz. Thus, (Y, 7) is cwN as claimed.

Now we construct another basis B for the topology 7. The basis B consists of the sets {(n, a ) } such that n E w and a E Ud and the sets

B A , a , n = { Z A } U {(n, a ) } U ((7% b ) : m > n A b E A}

for each triple (A ,a ,n ) such that A E A, a E A, and n E w . Applying cwNH(B) to this basis and the discrete set { X A : A E A} gives for each A E A a unique set

0 B A , ~ , ~ . The function f defined by f ( A ) = a is a choice function for A. T h e o r e m 7. The following are equivalent:

(i) MC. (ii) Every space ( X , T ) which is cwH, is also cwH(B) for a n y basis B closed under

(iii) The disjoint union of cwH spaces is cwH(B) for any basis B closed under finite

(iv) Every space ( X , T ) which is cwN, is also cwH(B) for any basis B closed under

P r o o f . The proof of (i)*(ii) is similar to the proof of (i)+(ii) in Theorem 6. (Here v would be a multiple choice function, and V = { n v ( i ) : i E k} B is the required family of open sets.) The proofs that (ii)+(iii) and (iii)+(iv) are again straightforward.

(iv)*(i). Let A = {Ai : i E k} be a family of infinite pairwise disjoint sets and let X be as in Theorem 1 (i)+ MC. We claim that X is cwN. Indeed, let D = {d j : j E J } be a discrete collection of closed sets. Clearly, for every i E k either yi $ dj for all j E J , or if yi E d j * for some j * E J , j * is unique. Then (Ai n d j i < w for all j E J , j # j * , and l { j E J : A, n d j # 0)l < w . For every j E J , put

finite intersections.

intersections.

finite intersections.

Oj = U{Ai n dj : i E k, yi $ d j } U (IJ{AT\& : yi E d j , ZI # J'}).

Clearly 0 = {Uj : j E J } is a disjoint family separating the members of D. It is easy to see that V = { { y i } : i E k} is a discrete family of closed subsets of X.

If B is defined by B = {AT\a : a E [Ai]<"\{0},i E k} U {{z} : z E Ud}, then B is a basis for X closed under finite intersections and D = { y i : i E k} is a closed discrete subspace of X . It follows from cwH(B) that there is a pairwise disjoint set V = {K : i E k} C B such that for each i E k, yi E K . Putting F = {Fi = Ai\K : i E k}, we see that each Fi, i E k, is a finite non-empty subset

0 o f Ai, which proves MC.

4 Summary

We summarize our results in the following diagram. (The forms cwH(B), DUcwH(B), and cwNH(B) in the diagram are abbreviations for (ii), (iii), and (iv), respectively, in Theorem 6.)


AC 0

U U U

8 0 0

0 U 0

0 U 0

cwH(B) * cwNH(B) e DUcwH(B)

DUP * MC * DUcwNN

DUPFCS DUPN DUMET

DUcwH DUM DUcwH

DUcwN DUMN DUcwNN

We have shown that DUM does not imply w-MC or MC in ZF'. (See Example 3 above. DUM is true in M , but w-MC and MC are false.) However, we still have the following questions.

Q u e s t i o n s .

(i) Does DUM imply w-MC or MC in ZF ? (ii) Does DUMN imply DUM ?

References

VAN DOUWEN, E. K. , Horrors of topology without AC: A non normal orderable space. Proc. Amer. Math. SOC, 95 (1985), 101 - 105. GOOD, C., and I. J. TREE, Continuing horrors of topology without choice. Topology and its Applications 63 (1995), 79 - 90. GOOD, C., I. J . TREE, and S. WATSON, On Stone's theorem and the axiom of choice. Preprint 1996. HOWARD, P., K. KEREMEDIS, H. RUBIN, and J. E. RUBIN, Versions of normality and some weak forms of the axiom of choice. Math. Logic Quarterly 44 (1998), 367 - 382. HOWARD, P., and J. E. RUBIN, Weak forms of the Axiom of Choice. In Preparation. JECH, T . J . , The Axiom of Choice. North-Holland Publ. Comp., Amsterdam 1973. LEVY, A., Axioms of multiple choice. Fund. Math. 50 (1962), 475 - 483. LO;, J. , and C. RYLL-NARDZEWSKI, Effectiveness of the representation theory for Boolean algebras. Fund. Math. 41 (1954), 49 - 56. PINCUS, D., Adding dependent choice. Annals Math. Logic 11 (1977), 105 - 145. SORGENFREY, R. H., On the topological product of paracompact spaces. Bull. Amer. Math. SOC. 53 (1947), 631 - 632. STEEN, L. A., and J . A. SEEBACH, Jr., Counterexamples in Topology. Springer-Verlag, Berlin-Heidelberg-New York 1986. WILLARD, S., General Topology. Addison-Wesley Publ. Go., Reading, MA, 1968.

(Received: June 18, 1997)

Disjoint Unions of Topological Spaces and Choice

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