Separable (and Metrizable) Infinite Dimensional Quotients of \(C_p(X)\) and \(C_c(X)\) Spaces

The famous Rosenthal-Lacey theorem states that for each infinite compact set K the Banach space C(K) of continuous real-valued functions on a compact space K admits a quotient which is either an isomorphic copy of c or \(\ell _{2}\). Whether C(K) admits an infinite dimensional separable (or even metrizable) Hausdorff quotient when the uniform topology of C(K) is replaced by the pointwise topology remains as an open question. The present survey paper gathers several results concerning this question for the space \(C_{p}(K)\) of continuous real-valued functions endowed with the pointwise topology. Among others, that \(C_{p}(K)\) has an infinite dimensional separable quotient for any compact space K containing a copy of \(\beta \mathbb {N}\). Consequently, this result reduces the above question to the case when K is a Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of \(\beta \mathbb {N}\)). On the other hand, although it is unknown if Efimov spaces exist in ZFC, we note under \(\lozenge \) (applying some result due to R. de la Vega), that for some Efimov space K the space \(C_{p}(K)\) has an infinite dimensional (even metrizable) separable quotient. The last part discusses the so-called Josefson–Nissenzweig property for spaces \(C_{p}(K)\), introduced recently in [3], and its relation with the separable quotient problem for spaces \(C_{p}(K)\).