Embeddings and frames of function spaces

For every Tychonoff space X we denote by C p ( X ) the set of all continuous real-valued functions on X with the pointwise convergence topology, i.e., the topology of subspace of R X . A set P is a frame for the space C p ( X ) if C p ( X ) ⊂ P ⊂ R X . We prove that if C p ( X ) embeds in a σ-compact space of countable tightness then X is countable. This shows that it is natural to study when C p ( X ) has a frame of countable tightness with some compactness-like property. We prove, among other things, that if X is compact and the space C p ( X ) has a Lindelöf frame of countable tightness then t ( X ) ⩽ ω . We give some generalizations of this result for the case of frames as well as for embeddings of C p ( X ) in arbitrary spaces.