Compactness of Powers of ω

Abstract

Abstract. We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary estensions. We also have results involving products of possibly uncountable regular cardinals.Mycielski [My], extending previous results by Ehrenfeucht, Erdös, Hajnal, Loś and Stone, showed that ω κ is not (finally) κ-compact, for every infinite cardinal κ strictly less than the first weakly inaccessible cardinal. Here ω denotes a countable topological space with the discrete topology; products (and powers) are endowed with the Tychonoff topology, and a topological space is said to be finally κ-compact if any open cover has a subcover of cardinality strictly less than κ.On the other direction, Mrówka [Mr1,Mr2] showed that if L ω 1 ,ω is pκ, κq-compact, then ω κ is indeed finally κ-compact (in particular, this holds if κ is weakly compact). As usual, L λ,µ is the infinitary language which allows conjunctions and disjunctions of ă λ formulas, and universal or existential quantification over ă µ variables; pκ, κq-compactness means that any κ-satisfiable set of |κ|-many sentences is satisfiable.To the best of our knowledge, the gap between Mycielski's and Mrówka's results has never been exactly filled. It follows from [Mr2, Theorem 1] anď Cudnovskiȋ [Ču, Theorem 2] that L κ,ω is pκ, κq-compact if and only if every product of |κ|-many discrete spaces, each of cardinality ă κ, is finally κ-compact (the proofs build also on work by Hanf, Keisler, Monk, Scott, Tarski, Ulam and others; earlier versions and variants were known under inaccessibility conditions). No matter how satisfying the above result is, it adds nothing about powers of ω, since it deals with possibly uncountable factors.In this note we show that Mrówka gives the exact estimation, namely, that ω κ is finally κ-compact if and only if L ω 1 ,ω is pκ, κq-compact. More 2010 Mathematics Subject Classification. Primary 54B10, 54D20, 03C75; Secondary 03C20, 03E05, 54A20, 54A25.

A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

“…Examples of subsets H of the set κ κ which are κ-compact subsets of the product space ( κ κ, τ pr ) but are not K κ subsets of the κ-Baire space, even when I − (κ) is not assumed, include by [15, Lemma 2.2] (and Tychonoff's theorem) the set H = κ 2 and, more generally, sets of the form H = α<κ I α , where I α ∈ [κ] <κ and I α is finite except for < κ many α < κ. (See [14] and the references therein for when this last assumption can be weakened.) Remark 2.11.…”

10.4115/jla.v6i0.205

“…To the best of our knowledge, the gap between the results by Mycielski and Mrówka mentioned above had not been exactly filled until we showed in [16] that Mrówka gives the optimal bound, that is, for κ regular, ω κ is finally κ-compact if and only if L ω 1 ,ω is (κ, κ)-compact. The aim of the present note is to show that the result holds also for a singular cardinal κ.…”

“…In order to generalize the above theorem (as well as other results proved in [16]) to the case when λ is singular we need a variant of the principle introduced in Definition 2.1. The modified principle deals with regularity of ultrafilters rather than with uniformity.…”

“…The principle we shall use here and which allows the possibility of dealing with singular cardinals is technically involved. As a sort of an introduction to it, we first recall the relatively simpler principle used in Lipparini [16], which is suitable for dealing with regular cardinals.…”

“…In Section 2 we establish notations, we introduce our main principle (λ, λ) ⇒(µ γ ) γ∈κ and we state the more general form of our results about powers of ω . We also make a comparison with the simpler situation described in [16], where we considered only the case when λ is regular. Section 3 deals with topology only; we provide a characterization of the principle (λ, λ) ⇒(µ γ ) γ∈κ and we present some applications to products of cardinals.…”