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2 TOPOLOGY PROCEEDINGS Volume 25, Summer 2000 Pages REFLECTING ON COMPACT SPACES Franklin D. Tall Abstract We consider whether, if a topological space reflects via an elementary submodel to a generalized Cantor discontinuum it must in fact equal its reflection. The answers involve large cardinals. Given an elementary submodel M of some H(θ) (see [JW, Chapter 24] for a careful elucidation of the implications of this) and a topological space X, T M, we define X M to be X M with topology generated by T M = {U M : U T M}. In [JT 1 ] we developed this notion; in [T 1 ] I proved that if X M is homeomorphic to the Cantor set, then X = X M. I. Juhász (personal communication) asked whether this generalized to arbitrary cardinals, i.e. if X M is homeomorphic to a generalized Cantor discontinuum D λ, where D is the 2-point discrete space, then does X = X M? We shall show that the answer is yes for small λ, but not necessarily for very huge ones. The following technical result is the key observation for our work on Juhász problem. It and Corollary 2 are due independently to Lucia Junqueira, conversations with whom have been very helpful. Note that when we write X M, we are implicitly assuming that X M. Also note that D λ M implies λ M. Research supported by NSERC Grant A Mathematics Subject Classification: Primary: 54D30; Secondary: 03E35, 03E55, 54B10. Key words: Elementary submodel, reflection, compact, generalized Cantor discontinuum, huge cardinals.

3 346 Franklin D. Tall Theorem 1. Let λ be a set. (Most of the time, it will be a cardinal.) Suppose (D λ ) M is compact. Then (D λ ) M is homeomorphic to D λ M. Proof. Let h :(D λ ) M D λ M be defined by h(f) =f (λ M). Claim h is a homeomorphism. Since (D λ ) M is compact and also T 2 (since D λ is [JT 1 ]), it suffices to show h is continuous, one-one, and has dense image. Let [p] ={g D λ M : g dom(p) =p}, where p is a finite partial function from λ M into D. Then h 1 ([p]) = {f D λ M : f dom(p) =p}. But this is open in (D λ ) M. h is one-one, since if f 1 f 2 are in D λ M, f 1 (λ M) f 2 (λ M) by elementarity. Finally, given any non-empty basic open [p] ind λ M, since dom(p) M, p M, so the function f defined by i) f dom(p) =p, ii) f ( λ dom(p) ) =0, is in (D λ ) M, and h(f) [p]. Corollary 2. If λ M and (D λ ) M is compact, then (D λ ) M = D λ. Proof. In this case, h is the identity function. Corollary 3. Let µ be the least ordinal not included in M. Then if λ is a cardinal less than µ and (D λ ) M is compact, then D λ =(D λ ) M. The proof is immediate. Corollary 4. Let µ be the least ordinal not included in M. If X M is homeomorphic to D λ, λ < µ, and D λ M, then X = X M. Proof. By [J], since X M is compact, so is X and there is a continuous map from X onto X M. Relativizing, there is a continuous map from X M onto (D λ ) M. Hence (D λ ) M is compact,

4 REFLECTING ON COMPACT SPACES 347 so D λ and hence 2 λ M. Therefore λ + M. We now do some easy calculation of cardinal functions. See [H] for definitions and theorems. Using a straightforward argument done in detail in [T 1 ], we see that X has no right- or left-separated subspaces of size λ +, else X M would. But w(x M )=λ. Since X M and hence [T 1 ] X is T 3, it follows that X T 2 λ,sox T M, so X = X M. Theorem 5. The first cardinal λ if any such that (D λ ) M is compact for some M but D λ must be strongly inaccessible. Proof. The first cardinal if any for which (D λ ) M is compact but D λ cannot be 2 κ for some κ<λ, κ M. By elementarity, we can omit κ M. The point is that since D κ is a continuous image of D λ (D λ ) M compact implies (D κ ) M is compact implies D κ =(D κ ) M implies 2 κ M implies λ M implies (D λ ) M = D λ. The first such cardinal can also not be singular, since D λ M implies λ M implies cf(λ) M implies D cf(λ) M (since D λ M). Then, since λ is least and as before (D cf(λ) ) M is compact, (D cf(λ) ) M = D cf(λ). Therefore D cf(λ) and a fortiori cf(λ) M. But then there is a set S of cardinals cofinal in λ included in M. For each σ S, D σ M and (D σ ) M is compact, so D σ =(D σ ) M so D σ and hence σ M. But then λ M and so D λ =(D λ ) M, contradiction. Thus M thinks λ is strongly inaccessible, so it is. Corollary 6. Suppose X M is homeomorphic to D λ M and λ is less than the first strongly inaccessible cardinal. Then X = X M. Proof. As for Corollary 4 above. Thus if there are no strongly inaccessible cardinals, Juhasz problem is solved. A less draconian solution is given by the following two results. 0 # is a set of natural numbers, the existence of which has large cardinal strength. See [K]. V = L implies 0 # does not exist.

5 348 Franklin D. Tall Corollary 7. If 0 # does not exist and M λ and (D λ ) M is compact, then (D λ ) M = D λ. Proof. This follows immediately from Lemma 8. [KT] If 0 # does not exist and M λ, then M λ. Corollary 9. If 0 # does not exist and X M is homeomorphic to D λ M, then X = X M. Proof. M X M =2 λ,so2 λ M, so as in the proof of Corollary 4, X = X M. By going to very large cardinals, we can find a λ such that (D λ ) M is compact but not equal to D λ. Definition. A cardinal λ is η-extendible if there is a ζ and an elementary embedding j : V λ+η V ζ, with critical point λ. See [K] to find out about such cardinals and about 2-huge ones, which we shall shortly introduce. Here we shall only mention that η-extendible cardinals are weaker in consistency strength than supercompact cardinals. Observe that for η 1, D j(λ) j V λ+η = {j(s) :j(s) D j(λ) and S V λ+η } = {j(s) :S D λ } = j D λ. Now if we want D j(λ) j V λ+η, we need η 2, for then D λ V λ+η,soj(d λ )=D j(λ) j V λ+η. We would be done if our definition of X M used V θ instead of H(θ) since j V λ+η is an elementary submodel of V ζ. To get H(θ), we use the fact that for inaccessible θ, V θ = H(θ), and work with a larger cardinal. Definition. λ is 2-huge if there is an elementary embedding j : V N, an inner model, with critical point λ such that j(j(λ)) N N.

6 REFLECTING ON COMPACT SPACES hugeness has considerably more consistency strength than supercompactness and assures us that j V j(λ) N, as is j D λ. j V j(λ) is an elementary submodel of V j(j(λ)) = H(j(j(λ))) (since j(j(λ)) is inaccessible by elementarity). As before, D j(λ) j V j(λ) and D j(λ) j V j(λ) = j D λ, which is compact T 2. (D j(λ) ) j Vj(λ) is also a T 2 (since D j(λ) is and T 2 goes down [JT 1 ]) topology on D j(λ) j V j(λ) that is weaker than the subspace topology and hence the two topologies are equal by compactness. Both j V j(λ) and V j(j(λ)) are in N; the proof that the former is an elementary submodel of the latter can be carried out in N. Thus, N thinks there is an elementary submodel M of H(j(j(λ))) such that (D j(λ) ) M is compact T 2 but D j(λ) (since j(λ) >λ). By elementarity, in V there is an elementary submodel M of H j(λ) such that (D λ ) M is compact T 2 but D λ. We have proved Theorem 8. If λ is 2-huge, then there is an elementary submodel M such that (D λ ) M is compact but D λ. There are several problems that remain: What is the consistency strength of the existence of a λ such that (D λ ) M is compact but D λ? Could such a λ be a successor cardinal? Must the first such λ be larger than merely strongly inaccessible? After this paper was completed, Lucia Junqueira [JT 2 ] proved that the condition that M λ can be removed from Corollary 7. It follows that the existence of a compact (D λ ) M D λ has consistency strength at least equal to the existence of 0 #. In [JT 2 ] we discuss in general when X M compact implies X M = X. In [T 2 ] we investigate the particular case of when X M is a dyadic compactum; the results obtained generalize those in this paper. Of course there are simple examples in ZFC of X s which are not equal to X M, even if the latter is compact. For example, let X be the one-point compactification of an uncountable discrete space and let M be countable.

7 350 Franklin D. Tall In [Ku], K. Kunen considerably sharpened the large cardinal bounds of Theorems 5 and 8. References [H] R. Hodel, Cardinal Functions I, 1-61, eds., K. Kunen and J.E. Vaughan, The Handbook of Set-theoretic Topology, North- Holland, Amsterdam, [J] L.R. Junqueira, Upwards preservation by elementary submodels, Top. Proc. 25, Spring [JT 1 ] L.R. Junqueira and F.D. Tall, The topology of elementary submodels, Top. Appl. 82 (1998), [JT 2 ], More reflections on compactness, submitted. [JW] W. Just and M. Weese, Discovering modern set theory, II in Graduate Studies in Mathematics, American Mathematical Society [K] A. Kanamori, The Higher Infinite, Springer Verlag, Berlin [Ku] K. Kunen, Compact spaces, compact cardinals, and elementary submodels, preprint. [KT] K. Kunen and F.D. Tall, The real line in elementary submodels of set theory, J. Symbolic Logic 65 (2000), [T 1 ] F.D. Tall, If it looks and smells like the reals,..., Fund. Math. 163 (2000), [T 2 ] F.D. Tall, Reflections on dyadic compacta, Top. Appl., to appear. Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3 CANADA address: tall@math.utoronto.ca

2 RENATA GRUNBERG A. PRADO AND FRANKLIN D. TALL 1 We thank the referee for a number of useful comments. We need the following result: Theorem 0.1. [2]

2 RENATA GRUNBERG A. PRADO AND FRANKLIN D. TALL 1 We thank the referee for a number of useful comments. We need the following result: Theorem 0.1. [2] CHARACTERIZING! 1 AND THE LONG LINE BY THEIR TOPOLOGICAL ELEMENTARY REFLECTIONS RENATA GRUNBERG A. PRADO AND FRANKLIN D. TALL 1 Abstract. Given a topological space hx; T i 2 M; an elementary submodel of