Injections into function spaces over ordinals

We investigate when and how function spaces over subspaces of ordinals admit continuous injections into each other. To formulate our results let τ be an uncountable regular cardinal. We prove, in particular, that: (1) If A and B are disjoint stationary subsets of τ then C p ( A ) does not admit a continuous injection into C p ( B ) ; (2) For A ⊂ ω 1 , C p ω ( A ) admits a continuous injection into C p ω ( ω 1 ) iff A is countable or ω 1 embeds into A (which, in its turn, is equivalent to the statement “ C p ω ( A ) embeds into C p ω ( ω 1 ) ”).