Eurozone’s Lemma

David Davis has proposed two geopolitical ideas:

  • For Northern Ireland to have dual EU/UK status;
  • For there to be a 10-mile ‘trade buffer zone’ between Northern Ireland and the Republic of Ireland.

The second is more interesting from a mathematical perspective: the 10-mile buffer zone means that (the closures of) Northern Ireland and the Republic of Ireland are disjoint compact subsets of a normal topological space. By Urysohn’s Lemma, this means that there exists a continuous function f : Ireland \rightarrow [0, 1] such that f is identically 0 on Northern Ireland and identically 1 on the Republic of Ireland.

The proof of this proceeds as follows:

  • By taking closures, assume without loss of generality that NI and ROI are both closed and disjoint (the interior 10-mile buffer zone is not considered to belong to either).
  • Define U(1) and V(0) to be the complements of NI and ROI, respectively. These are overlapping open sets, whose intersection is the buffer zone.
  • For each k \in \{1, 2, 3, \dots \}:
    • For each dyadic rational r \in (0, 1) with denominator 2^k and odd numerator:
      • Let q = r - 2^{-k} and s = r + 2^{-k}, so q,r,s are adjacent;
      • By appealing to the normality of Ireland, let U(r) and V(r) be two disjoint open sets containing the complements of V(q) and U(s), respectively.
  • Now we have disjoint open sets U(r) and V(r) for each dyadic rational r, such that the U(r) form an ascending chain of nested spaces.
  • Define f(x) := \inf \{ r : x \in U(r) \} (where the infimum of an empty set is taken to be 1).

With this interpolating function f, it is easy to take convex combinations of EU and UK standards. For example, a road sign at a point x must be stated in ‘lengths per hour’, where one length is exactly 1 + 0.609344(1 – f(x)) kilometres.

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