A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications

We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff $X$ equipped with topology pointwise convergence is distinguished if and only $\Delta$-space in sense \cite {Knight}. As an application this characterization theorem we obtain following results: 1) If \v{C}ech-complete (in particular, compact) such $C_p(X)$ distinguished, then scattered. 2) For every separable compact Isbell--Mr\'owka type $X$, distinguished. 3) ordinals $[0,\omega_1]$, not observe existence uncountable metrizable independent ZFC. explore also question to which extent class $\Delta$-spaces invariant under basic topological operations.