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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