Injections into function spaces over compacta

摘要

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 r 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.