Abstract:By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the “reduced coproduct”, which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the “ultracoproduct” can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealin… Show more
“…Also, because the ultracoproduct operation on subsets commutes with finite unions and intersections, and because (see Proposition 1.5 in [1]) the Boolean lattice of clopen subsets of an ultracoproduct of compacta is the corresponding ultraproduct of the clopen set lattices of those compacta, we infer that…”
Section: Applications To Multicoherence Degree In Continuamentioning
confidence: 95%
“…Given x i : i ∈ I ∈ i∈I X i , there is just one point of D X i containing D {x i } as an element; call this point D x i . Then, by basic results in [1], the set of such points is dense in D X i . In light of this, we fix D x i ∈ H \ K and D y i ∈ K \ H.…”
Section: Applications To Multicoherence Degree In Continuamentioning
confidence: 96%
“…Proof. By Theorem 2.6 in [5], the class of compacta of covering dimension ≤ n is closed under co-existential maps; by Theorem 2.2.2 in [1], it is closed under ultracopowers. A nearly identical argument shows the class to be closed under all ultracoproducts.…”
Section: Applications To Dimensionmentioning
confidence: 99%
“…What makes Theorems 2.6 in [5] and 2.2.2 in [1] work is the theorem of E. Hemmingsen (Lemma 2.2, and its corollary, in [9]) to the effect that a normal Hausdorff space X has covering dimension ≤ n if and only if, whenever {B 1 , . .…”
Section: Applications To Dimensionmentioning
confidence: 99%
“…The most flexible way to form D X i for our present purposes is to take the following steps (see [1,3]):…”
Section: Introduction and The Main Theoremmentioning
Abstract. The Chang-Loś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications.
“…Also, because the ultracoproduct operation on subsets commutes with finite unions and intersections, and because (see Proposition 1.5 in [1]) the Boolean lattice of clopen subsets of an ultracoproduct of compacta is the corresponding ultraproduct of the clopen set lattices of those compacta, we infer that…”
Section: Applications To Multicoherence Degree In Continuamentioning
confidence: 95%
“…Given x i : i ∈ I ∈ i∈I X i , there is just one point of D X i containing D {x i } as an element; call this point D x i . Then, by basic results in [1], the set of such points is dense in D X i . In light of this, we fix D x i ∈ H \ K and D y i ∈ K \ H.…”
Section: Applications To Multicoherence Degree In Continuamentioning
confidence: 96%
“…Proof. By Theorem 2.6 in [5], the class of compacta of covering dimension ≤ n is closed under co-existential maps; by Theorem 2.2.2 in [1], it is closed under ultracopowers. A nearly identical argument shows the class to be closed under all ultracoproducts.…”
Section: Applications To Dimensionmentioning
confidence: 99%
“…What makes Theorems 2.6 in [5] and 2.2.2 in [1] work is the theorem of E. Hemmingsen (Lemma 2.2, and its corollary, in [9]) to the effect that a normal Hausdorff space X has covering dimension ≤ n if and only if, whenever {B 1 , . .…”
Section: Applications To Dimensionmentioning
confidence: 99%
“…The most flexible way to form D X i for our present purposes is to take the following steps (see [1,3]):…”
Section: Introduction and The Main Theoremmentioning
Abstract. The Chang-Loś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications.
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