Here are two kind-of dual results that have now been proved after I asked them here:

The first arose from Induced map on path manifolds: is it a submersion?, which got an answer in very general terms involving generalised smooth spaces in this preprint:

- Andrew Stacey,
*Yet More Smooth Mapping Spaces and Their Smoothly Local Properties*, arXiv:1301.5493.

and then again in this paper, as Lemma 2.4:

- Habib Amiri, Alexander Schmeding
*A differentiable monoid of smooth maps on Lie groupoids*, Journal of Lie Theory **29** (2019), No. 4, 1167–1192, arXiv:1706.04816

in a way that only partially overlaps with Andrew Stacey's version; in one sense it's less general, but it seems to use a stronger notion of submersion/regular map.

The second came from Extension of functions from geodesically convex compact sets in a Riemannian manifold, which has now been answered in

- David Michael Roberts, Alexander Schmeding,
*Extending Whitney's extension theorem: nonlinear function spaces*, to appear, Annales de l'Institut Fourier, arXiv:1801.04126

A different kind of question I asked now has three papers giving three different approaches, namely On a weak choice principle, which led to (in chronological order):

Benno van den Berg, *WISC is independent of ZF*, (pdf), also Theorem 5.1/Corollary 5.2 in *Predicative toposes*, arXiv:1207.0959. This uses Gitik's class forcing symmetric model of ZF, over ZFC with a reasonably strong large cardinal assumption.

Asaf Karagila, *Embedding Orders Into Cardinals With $DC_\kappa$*, Fund. Math. **226** (2014), 143-156, doi:10.4064/fm226-2-4, arXiv:1212.4396. This uses class forcing symmetric models, over ZFC with no large cardinals.

David Michael Roberts, *The weak choice principle WISC may fail in the category of sets*, Studia Logica Volume **103** (2015) Issue 5, pp 1005-1017, doi:10.1007/s11225-015-9603-6 arXiv:1311.3074. This uses topos-theoretic methods, over a well-pointed base topos with no Choice.

Some people might feel uneasy citing their own work at a thread entitled "Best of MathOverflow", but perhaps that title should be interpreted broadly$\endgroup$ – Martin Sleziak Jan 21 '18 at 9:49