The homotopy classes of continuous maps between some nonmetrizable manifolds

We prove that the homotopy classes of continuous maps Rn→R, where R is Alexandroff's long ray, are in bijection with the antichains of P({1,…,n})\{∅}. The proof uses partition properties of continuous maps Rn→R. We also provide a description of [X,R] for some other nonmetrizable manifolds.