Separable quotients in 𝐶_{𝑐}(𝑋), 𝐶_{𝑝}(𝑋), and their duals

Abstract

The quotient problem has a positive solution for the weak and strong duals of C c ( X ) C_{c}\left ( X\right ) ( X X an infinite Tichonov space), for Banach spaces C c ( X ) C_{c}\left ( X\right ) , and even for barrelled C c ( X ) C_{c}\left ( X\right ) , but not for barrelled spaces in general. The solution is unknown for general C c ( X ) C_{c}\left ( X\right ) . A locally convex space is properly separable if it has a proper dense ℵ 0 \aleph _{0} -dimensional subspace. For C c ( X ) C_{c}\left ( X\right ) quotients, properly separable coincides with infinite-dimensional separable. C c ( X ) C_{c}\left ( X\right ) has a properly separable algebra quotient if X X has a compact denumerable set. Relaxing compact to closed, we obtain the converse as well; likewise for C p ( X ) C_{p}\left ( X\right ) . And the weak dual of C p ( X ) C_{p}\left ( X\right ) , which always has an ℵ 0 \aleph _{0} -dimensional quotient, has no properly separable quotient when X X is a P-space of a certain special form X = X κ X=X_\kappa