Koch’s arc theorem

Koch’s arc theorem is a famous theorem saying that, under a few conditions, given any point x of an open subset U of a pospace (X, ≤), one can draw an arc chain from some point on the boundary of U to x. The conditions are that: (i) the closure of U should be compact and (ii) U should not contain any local minimum. That may seem clear: starting from x, go down inside U, and continue forever; this cannot stop while in U because U has no local minimum, hence must go all the way to its boundary. The actual proof requires quite some ingenuity, and builds the desired arc chain as a maximal chain for an ordering that is itself minimal in some well-crafted family, using Zorn’s Lemma twice. Read the full post.