The homotopy classes of continuous maps between some nonmetrizable manifolds

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