(i) GCH - American Mathematical Society

proceedings of the american mathematical society Volume 118, Number 4, August 1993

THE a-BOUNDIFICATION OF a SALVADORGARClA-FERREIRAAND ANGEL TAMARIZ-MASCARUA (Communicated by Franklin D. Tall)

Abstract. A space X is < a-bounded if for all A C X with |^| for every singular cardinal.

1. Preliminaries Throughout this paper, all spaces are assumed to be completely regular and Hausdorff. If A is a space and B C X, &x B denotes the closure of B in X. For x £ X, JV(x) is the set of neighborhoods of x in X. £P(X) is the set of all subsets of a set X . The Greek letters stand for ordinal numbers; in particular, a, k , 6 denote infinite cardinal numbers; y, v, p denote arbitrary cardinals; and 8, £, X, n denote ordinal numbers. For a cardinal a, we let a+ stand for the smallest cardinal greater than a. For k , y cardinals we set [K]y = {M C k : \M\ = y} and [k]Y is a continuous function, we let /: PX —>PY stand for the Stone extension of /. The remainder of PX is X* = pX\X;

in particular,

a* = P(a)\a.

For A C a we have that

(see [2, Chapter 2]) A = {p £ P(a) : A £ p} = clB{n]A and

A* = A\A .

We shall use the terminology and notation of Comfort and Negrepontis [2]. The notion of a-bounded space was introduced in [7] and modified by Com-

fort as follows. 1.1. Definition. A space X is < a-bounded if for every ACX less than a, dx A is a compact set.

of cardinality

It is evident that every space is < cobounded, and if X is < a-bounded. then X is < y-bounded for every y < a. The basic properties of < a-bounded spaces are summarized in the following proposition (see, e.g., [7, 8]). 1.2.

Proposition. Let a be a cardinal number. Then (a) every compact space is < a-bounded; (b) every closed subset of a < a-bounded space is < a-bounded; (c) the product of a set of < a-bounded spaces is < a-bounded; (d) the intersection of a set of < a-bounded spaces is < a-bounded; (e) the continuous image of a < a-bounded space is < a-bounded.

Notice that (d) is a particular case of Lemma 2 of [8], and it is a consequence

of (b) and (c). 1.3. For a < a-bounded space Z and X c Z , we set Bn(X, Z) = f]{Y : X C Y c Z and Y is < a-bounded}. It follows from 1.2(d) that Bn(X, Z) is the smallest < a-bounded space containing X and is contained in Z . If Z = px , then B„(X, Z) will be denoted by Ba(X). In this case B„(X) has the following extension property: For each

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THE cr-BOUNDIFICATIONOF a

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< a-bounded space Y and each continuous function f:X—*Y, there exists a continuous function /: Ba(X) -> Y such that f\x = f [5, 8]. Ba(X) is called

the a-boundification of X. 1.4. Notation. For a space A, we define B°a(X) = X;

Bi+x(X) = lJ{cV^l

: A C Bi(X) and |4| < a} for each ordinal

£ < a+ ; and

*£W = [Jx (e) are trivial, and (a) => (f), (a) =>■(d), and (e) => (c) are direct consequences of Lemmas 2.8 and 2.9. In order to complete the proof we will show (a) ■&(b) and (c) => (a). (a) => (b) Suppose that a is singular and B(a) is locally compact. Then B(a) n U(a) is a nonempty open subset of U(a). Fix an arbitrary p £ U(a). We will show that p £ B(a). Indeed, since the type T(p) = {q £ P(a): there is a permutation ft of a such that h(p) = q} of p is a dense subset of U(a) (see [2] for a proof), there is a £ T(p) n B(a) n U(a). Choose a permutation / of a such that f(q) = p . According to Proposition 2.6, we have that B(a) is 7?(a)-compact and so f(q) = p £ B(a); thus, p £ B(a). But this implies that P(a) = B(a), a contradiction to Corollary 2.4(b). (b) => (a) Suppose that a is a regular cardinal. From 0.1 (a) it follows that B(a) = N(a). Since N(a) is open in P(a), we have that B(a) is locally compact.

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THE c-BOUNDIFICATIONOF a

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(c) => (a) If a is a regular cardinal and £P = {cl^(Q)k : k < a} , then f is a cozero cover of N(a) = B(a) of cardinality a without a finite subcover. Now Retta's result (Lemma 2.8) implies that B(a) is not a-pseudocompact. 3. The sets B2(a) and

7?3(a)

Our main goal here is to give an answer to Comfort's question 0.2 in the affirmative when co = cf(a) < a (see 3.5, 3.7, and 3.8). Because of 0.1, we will only be concerned with singular cardinals. Thus, throughout this section, a will denote a singular cardinal.

3.1. Definition (Keisler [11]). For p £ X, let G(p) = minjy : y is a cardinal number and p is not y+-good}. G(p) is called the degree of goodness of p .

3.2. We point out that B2(a)\Bx(a) is dense in U(a). Indeed, let {k4 : £ < cf(a)} be a strictly increasing sequence of cardinals converging to a (k^ / a), and let A £ [a]a . Choose #* € WK((a)nA for each £ < cf(a). If p £ P(a) is a complete accumulation point of {p^:£ < cf(a)} , then p £ (B2(a)\Bx(a))nA . We will prove in 3.5 that, for each k < a, the subset of B2(a)\Bx(a) of all ultrafilters of degree of goodness equal to k+ is dense in U(a). For our purpose we need the following two lemmas (they are Theorems 10.5 and 10.6

of [2], respectively). 3.3. Lemma (Keisler [12]). Let co < y < k , let r be a function from k onto y, and let e: P(y) -» P(k) be a continuous function such that Toe is the identity function on P(y). If q £ P(y) is countably incomplete and p = e(q) £ P(k) , then p and q have the same degree of goodness. Let k be a cardinal number. A family &~ of subsets of k is said to have the uniform finite intersection property if & # 0 and | f\k a* by /(£)= P^j^) for £ f(q). Clearly, for every f £ aK , pf £ WK(a). Since {£ < k : cpj(£) = tf>j(£)}= {£
j(p)= cf>g(p)= pg if and only if / = g. Using Theorem 4.2, we have that a* = \aK/q\ < \WK(a)\.

The next result appears in [3] without proof. 4.5.

Corollary. For every a, the following equality holds: |A(a)| = a«*.£22\

Proof. We have that N(a) = [){Wy(a) : y < a}. that \N(a)\ = Zy a. (a) cf(|A(a)|) = cf(22")>2e>a. (b) If y is a cardinal less than a, then |A(a)|>' = 2(2'V;' = 22" = |A(a)|. 4.9. Theorem. Let co < a. Then \B(a)\ = \N(a)\ whenever a satisfies one of the following properties: (a) a is a regular cardinal. (b) a is a singular cardinal which is not a strong limit and sup,/ 22" for every v < a}. Observe that (see 4.6) if a is not a strong limit and {2r}y, we consider the function $$: cf(a) -» Bl(a) defined by cj>f(£)= piJ/( for £ < cf(a). Fix q £ U(cf(a)). Then 4>f(q) £ cl^(Q)cj)f(cf(a)) C B2(a). It suffices to prove that the relation f —> ~4>f(q) from |A(a)|cf(a)

to B2(a)

is one-to-one.

Indeed, let f,g£

|A(a)|cf(a>

such that f ^ g. It is evident that 4>f(£)= (j>g(£)iff Xf $ = Xg ^ and so \{£ < cf(a) : cpf(£)= J>g(£)}\= \{£ < cf(a) : XfA = kgf(£) = 4>g(£)} = |jB(Q)|< f^jA y