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 such that 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 :
  • For each dyadic rational with denominator and odd numerator:
    • Let and , so 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 (where the infimum of an empty set is taken to be 1).

With this interpolating function , 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.