On linear continuous operators between distinguished spaces $$C_p(X)$$

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, locally convex $$C_{p}(X)$$ is distinguished if only X $$\Delta $$ -space. If there exists linear continuous surjective mapping $$T:C_p(X) \rightarrow C_p(Y)$$ $$C_p(X)$$ distinguished, then $$C_p(Y)$$ also (Ka̧kol Proc B, 2021). Firstly, this paper we explore the following question: Under which conditions operator above open? Secondly, devote special attention to concrete spaces $$C_p([1,\alpha ])$$ , where $$\alpha countable ordinal number. A complete characterization of all Y admit $$T:C_p([1,\alpha ]) given. We observe that every closed subspaces are thereby answering an open question posed Using some properties -spaces prove surjection C_k(X)_w$$ $$C_k(X)_w$$ denotes Banach C(X) endowed with its weak topology, does not exist infinite metrizable compact C-space (in particular, $$X \subset {\mathbb {R}}^n$$ ).