Let $X$ be a compact metrizable topological semilattice with a neighbourhood base of sub-semilattices that are path connected. I need a reference for a proof that $X$ is an absolute retract.
Here is what I have:
- There is a proof in Theorem~IV-3.17 "The theory of convex structures" by M.L.J. van de Vel - 1993 http://books.google.com.au/books?isbn=0080933106 But there are a number of problems with this:
While the proof seems to be correct the statement of the theorem seems wrong, it forgets to mention that the abstract convex spaces is locally convex. You have to wade through abstract convexity definitions and their relation with semilattices to figure out stuff. Also, the book is super expensive online.
There is a proof outlined in Exercise 3.20 p. 297 of "A Compendium of Continuous Lattices" by Gierz, Hofmann, Keimel, Lawson, Mislove, Scott; http://www.mathematik.tu-darmstadt.de/~keimel/compend.ps.gz establishing this as a consequence of the Wojdyslawski (1937) result that the set of all non-empty closed subsets of a Peano continuum is an absolute retract. But it's an exercise. (I'm definitely not looking for anyone here to solve that exercise, it's straightforward)
There is even a proof in a very recent paper that effectively roles its own...