Injections into function spaces over compacta

We study the topology of X given that C p ( X ) injects into C p ( Y ) , where Y is compact. We first show that if C p over a GO-space injects into C p over a compactum, then the Dedekind remainder of the GO-space is hereditarily paracompact. Also, for each ordinal τ of uncountable cofinality, we construct a continuous bijection of C p ( τ , { 0 , 1 } ) onto a subgroup of C p ( τ + 1 , { 0 , 1 } ) , which is in addition a group isomorphism.