ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

upper bound). Such proofs are valid in Hθ for sufficiently large regular θ. Henceforth, θ
will denote a sufficiently large regular cardinal.
We will use elementary submodels of the {∈}-structure Hθ (with the symbol “∈”
interpreted as actual membership) to greatly simplify and shorten “closing off” argu-
ments that appear in many of our proofs. Sometimes arbitrary elementary submodels
of Hθ will not be sufficiently closed off for our purposes. One easy fix is to add constant
symbols for a small number of objects that we care about. For example, it sometimes
suffices simply to expand 〈Hθ, ∈〉 to 〈Hθ, ∈, C(X)〉 for some space X that we want our
elementary substructures to “know” about. When this trick does not suffice, we will use
elementary chains.
Definition 1.6.3. A sequence of models 〈Mα〉α