LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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The Läuchli Urysohn obstruction is injectively boundable

Statement

Let $T$ be the sentence asserting that there are a topological space $(X, τ)$ and disjoint closed sets $E_{0}, E_{1} \subseteq X$ such that $X$ is normal and no continuous $f : X \to R$ satisfies $f [E_{0}] \subseteq {0}$ and $f [E_{1}] \subseteq {1}$ . The sentence $T$ is boundable, hence injectively boundable, and it admits an atom-blind typed transfer certificate in the sense of Boundable sentences over an atom set. A fixed absolute bound below $ω + ω$ captures all subsets of $X$ , members of $τ$ , and candidate real-valued function graphs. Brunner's ordered continuum is a witness to $T$ in each of the two permutation models.

Facts & Assumptions

Given: The ordered Läuchli continuum $L$ of Brunner's ordered Läuchli permutation models, its two endpoint closed sets, and the failure of Urysohn's lemma in the two models of Brunner's models satisfy the required choice and Urysohn obstructions.

[F1]

Boundable sentences over an atom set: a formula is boundable only when it is provably equivalent, uniformly in ZFA, to its relativisation to $V_{α} (⋃ \vec{x})$ for a fixed absolutely defined ordinal $α$ ; a syntactic restriction alone is insufficient (Boundable sentences over an atom set).

[F2]

The space $L$ is a compact linearly ordered normal space with two distinct closed endpoint singletons, and every continuous real-valued function on it is constant (Brunner's models satisfy the required choice and Urysohn obstructions, Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly, Continuity of a map of topological spaces at a point and globally).

[F3]

Every boundable statement is, up to equivalence, injectively boundable (Pincus's Fact 5.4 as reproduced in the cited Tachtsis paper).

[L1]

For the parameter tuple $(X, τ, E_{0}, E_{1})$ put $B = X \cup τ \cup E_{0} \cup E_{1}$ . Then $X, τ, E_{0}, E_{1}$ and every subset of $X$ lie in $V_{1} (B)$ . The canonical pure codes for $R$ , its topology, $0$ and $1$ , and every graph $f \subseteq X \times R$ lie in $V_{ω + n} (B)$ for one fixed finite $n$ : ordered-pair and graph coding adds only finitely many power-set iterations. Hence all of them lie below $V_{ω + ω} (B)$ (Boundable sentences over an atom set).

Proof

technique · direct

1.1

Let $Φ (X, τ, E_{0}, E_{1})$ be the following fixed membership-language formula: $τ \subseteq P (X)$ is a topology; $E_{0}, E_{1} \subseteq X$ are disjoint, nonempty and closed; every two disjoint closed subsets $C, D \subseteq X$ are contained in disjoint members of $τ$ ; and there is no function graph $f \subseteq X \times R$ whose inverse image of every open subset of $R$ belongs to $τ$ and which takes the constant values $0$ on $E_{0}$ and $1$ on $E_{1}$ . This says exactly that $(X, τ)$ is normal and the specified closed pair has no Urysohn separator.

2.1

To establish boundability it suffices to prove the uniform ZFA equivalence between $Φ$ and its relativisation to the fixed segment $V_{ω + ω} (B)$ .

step 1.1F1suffices: uniform relativisation

2.2

Expand the abbreviations in step 1.1. The topology axioms quantify over members and subfamilies of $τ$ ; closedness and normality quantify over subsets of $X$ and members of $τ$ ; the function condition quantifies over ordered-pair graph entries; and continuity quantifies over the fixed pure real topology and subsets of $X$ obtained as graph preimages. Every one of these domains is contained in the relative segment of [L1]. Consequently ZFA proves $Φ (X, τ, E_{0}, E_{1}) ⟷ Φ^{V_{ω + ω} (B)} (X, τ, E_{0}, E_{1}) .$ For the forward implication, every quantified object in the expanded formula is present in the segment by step 1.1 and [L1], so restricting the quantifiers loses no candidate closed set, normality witness, real open set, or function graph. For the reverse implication the same domain equalities show that each restricted universal quantifier ranges over the entire bounded sort named in the unrestricted formula, and each restricted existential witness is an actual member of that sort. Thus the two formulas have identical bounded domains, uniformly in every ZFA universe.

step 1.1step 2.1L1F1

3.1

The relativised formula is atom-blind: its only atomic tests are equality and membership among the carried sorts and the fixed pure real codebook. Points of $X$ are treated opaquely; the formula never asks whether a point is an atom or examines any members it may have outside the carried incidence structure. Therefore the same typed formula describes a normal space and a failed separator after an atom-to-set embedding.

step 1.1step 2.2L1F1

4.1

By step 2.2 and [F1], the existential closure of $Φ$ is boundable with the fixed absolute bound $ω + ω$ ; by [F3] it is injectively boundable, and step 3.1 supplies the atom-blind typed certificate. In each Läuchli model, take $X = L$ , $τ$ its order topology and $E_{0} = {a}$ , $E_{1} = {b}$ . They satisfy $Φ$ by [F2], because a separator would be a nonconstant continuous real-valued map.

step 1.1step 2.2step 3.1F1F2F3discharge-construct∎

Remarks

Why a certificate is needed at all. The transfer theorem used below accepts a sentence together with an absolute bound and a typed incidence structure; without the certificate the transfer step would have to be taken on trust. The certificate produced here is the one the Pincus and Jech–Sochor interfaces consume.
What the certificate does not say. It speaks only of the carried continuum and its separating functions; it asserts nothing about the rest of the permutation model, and in particular it does not certify countable choice or BPI, which are transferred through the separate exceptional clauses.

Depends on

Used by

Dependency tree · two levels

33 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Norbert Brunner, Geordnete Läuchli Kontinuen (standard reference, not scraped)
Eleftherios Tachtsis, The Boolean prime ideal theorem does not imply the extension of almost disjoint families to MAD families (standard reference, not scraped)