ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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Finite-menu intersection in the DMC Urysohn construction

Example

In the DMC construction of DMC implies Urysohn's lemma each level $n$ of the dyadic scale is obtained by intersecting, coordinatewise, the finitely many nodes of the menu at that level. The example computes the first two levels and verifies that the closure inclusions survive the intersection and that the predecessor coherence of the menus is preserved.

Facts & Assumptions

Given: A normal space $X$ with disjoint closed sets $F, G$ ; nodes $⟨ U_{1}, \dots, U_{2^{n}} ⟩$ of open sets with $\overline{U_{i}} \subseteq U_{i + 1}$ for $1 \leq i < 2^{n}$ , $F \subseteq U_{1}$ and $U_{2^{n}} = X ∖ G$ ; and nonempty finite menus $M_{n}, M_{n + 1}$ of nodes of levels $n$ and $n + 1$ , respectively. The successor relation used here is the one fixed in DMC implies Urysohn's lemma: for $a \in M_{n}$ and $b \in M_{n + 1}$ , $a S b$ means $b_{2 i} = a_{i}$ for every $1 \leq i \leq 2^{n}$ . Every $a \in M_{n}$ has an $S$ -successor in $M_{n + 1}$ , and every $b \in M_{n + 1}$ has an $S$ -predecessor in $M_{n}$ .

[F1]

The closure of a finite union is the union of the closures, and for finitely many sets the closure of an intersection is contained in the intersection of the closures; $A \subseteq \overline{A}$ (Interior, closure, boundary, exterior, derived set and isolated point in a topological space, Interior commutes with finite intersections and closure with finite unions, while the two reverse combinations are inclusions only and both fail for infinite families; the space is the disjoint union of interior, boundary and exterior).

[F2]

A finite intersection of open sets is open, and the entries of a node satisfy the closure inclusions displayed above (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison, Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly).

[F3]

Abstract DMC supplies successor menus, without prescribing their levels (Dependent multiple choice in finite-level tree form). Here level alignment and both coverage properties are explicit Given hypotheses; their construction in DMC implies Urysohn's lemma is context, not an additional premise of this finite calculation. Natural numbers include zero (The natural numbers $N$ (von Neumann)).

Verification

1.1

Let $a^{(1)}, \dots, a^{(m)}$ be the nodes of the menu at level $n$ , with entries $a_{i}^{(j)}$ , and put $U_{i} : = ⋂_{1 \leq j \leq m} a_{i}^{(j)}$ for $1 \leq i \leq 2^{n}$ ; each $U_{i}$ is a finite intersection of open sets, hence open.

givenF2

2.1

For each $1 \leq i < 2^{n}$ the inclusion $\overline{U_{i}} \subseteq U_{i + 1}$ holds: $\overline{U_{i}}$ is contained in $⋂_{1 \leq j \leq m} \overline{a_{i}^{(j)}}$ by [F1], and each $\overline{a_{i}^{(j)}} \subseteq a_{i + 1}^{(j)}$ by hypothesis, so $\overline{U_{i}} \subseteq ⋂_{1 \leq j \leq m} a_{i + 1}^{(j)} = U_{i + 1}$ .

step 1.1F1

2.2

The boundary values are preserved: $F \subseteq U_{1}$ because $F$ is contained in every $a_{1}^{(j)}$ , and $U_{2^{n}} = X ∖ G$ because every $a_{2^{n}}^{(j)}$ equals $X ∖ G$ .

step 1.1F2

2.3

Predecessor coherence is preserved: the Given predecessor property and the definition of $S$ show that every $2 i$ -th entry occurring in the level- $(n + 1)$ menu is an $i$ -th entry occurring in the level- $n$ menu. Conversely, the successor property in the Given data makes every level- $n$ node occur as the predecessor of some level- $(n + 1)$ node, so every level- $n$ $i$ -th entry occurs among those $2 i$ -th entries. The two indexed families of sets therefore have the same range, and their intersections are equal.

step 1.1given

3.1

At level $0$ the only possible node is $⟨ X ∖ G ⟩$ , so its intersection is $X ∖ G$ . A finite menu at level $1$ consists of nodes $⟨ V_{j}, X ∖ G ⟩$ , $1 \leq j \leq m$ , with $F \subseteq V_{j}$ and $\overline{V_{j}} \subseteq X ∖ G$ . Its two intersections are $V = ⋂_{1 \leq j \leq m} V_{j}$ and $X ∖ G$ . Steps 2.1 and 2.2 give $F \subseteq V \subseteq \overline{V} \subseteq X ∖ G$ ; the second intersection equals the level- $0$ value as in step 2.3. When $m = 1$ these are just the original entries. All finite enumerations here concern one fixed menu; no sequence of enumerations is selected.

step 2.1step 2.2step 2.3givenF1F3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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

alg-d, Urysohn no hodai (Urysohn's lemma) (standard reference, not scraped)