Euler-Poincar\'e obstruction for pretzels with long tentacles \`a la Cantor-Nyikos

We present an avatar of the Euler obstruction to foliated structures on certain non-metric surfaces. This adumbrates (at least for the simplest 2D-configurations) that the standard mechanism—to the effect that the devil of algebra sometimes barricades the existence of angelic geometric structures (obstruction theory more-or-less)—propagates slightly beyond the usual metrical proviso. Alas, the game is much more conservative than revolutionary: in particular we enjoyed retrospecting at Poincar'e’s argument of 1885 (announced in 1881).