A Construction in Set‐Theoretic Topology by Means of Elementary Substructures

Universal flows and automorphisms of P(ω)/fin

“…This definition is essentially due to Bandlow; see [2]. It is not immediately clear that the function f M is well-defined, but this turns out to be a consequence of elementarity, articulated in the following lemma.…”

“…Every trivial automorphism embeds in its inverse. In this subsection, we will break briefly from classifying the universal trivial automorphisms to observe an interesting consequence of Theorem 3.7 (2).…”

“…Proof. To prove (1) implies (2), first note that if (1) holds, then the definition of a U -limit implies (using the fact that U is non-principal) that both U -lim n∈D y n and U -lim n∈D z n are in X. Let y = U -lim n∈D y n and z = f (y), and let V be an open neighborhood of z (in Y ).…”

“…x n = π(ϕ * p (U )) = π • ϕ * p (U ) (Lemma 4.6 was applied to get the third equality, and Lemma 4.1 (2) to get the fourth).…”

The (λ,κ)-Freese-Nation Property for Boolean Algebras and Compacta

“…In [5], Heindorf proved that a boolean algebra has the FN iff it is openly generated (a property we need not define here), and in [9] and [10],Ščepin studied the Stone dual of open generation and generalized it from the Stone spaces to all compacta (i.e., compact Hausdorff spaces).Ščepin proved that the k-adic compacta, which are the continuous images of openly generated compacta, are a superclass of the dyadic compacta (i.e., the continuous images of powers of 2) and that the k-adic compacta and dyadic compacta satisfy the same major structural theorems and cardinal function equations. Bandlow [1] translatedŠčepin's definition into the language of elementary substructures: a compactum X is openly generated iff, for a club of countable elementary submodels M of (H(θ), ∈, T X ), the quotient map π X M : X → X/M is open. Here π X M is defined by declaring π X M (p) = π X M (q) iff there are disjoint closed neighborhoods U of p and V of q such that U, V ∈ M .…”