For a Tychonoff space X by \(C_p(X)\) and \(C_k(X)\) we denote the space C(X) of continuous real valued functions on X endowed with the pointwise topology \(\tau _{p}\) and the compact-open topology \(\tau _{k}\) , respectively. If X is a pseudocompact space, then the uniform topology \(\tau _{\infty }\) on C(X) is generated by the sup norm \(\Vert \cdot \Vert _{\infty }\) ; clearly \(C_{\infty }(X)=(C(X), \tau _{\infty })=(C(X), \Vert \cdot \Vert _{\infty })\) is a Banach space. We characterize \(\omega \) -bounded (hence also compact) scattered spaces in terms of \(C_{\infty }(X)\) and \(C_p(X)\) (supplementing results characterizing compact scattered spaces X due to Namioka-Phelps, Gerlits, Pytkeev, Pełczyński-Semadeni, Lotz-Peck-Porta and Ka̧kol-Kurka). Applications yielding a \(C_p\) -version of the Pełczyński-Semadeni theorem are studied. We prove (among other results): A compact space X is scattered if and only if every infinite-dimensional \(\tau _{\infty }\) -closed subspace of \(C_p(X)\) contains an isomorphic copy of \(c_0\) (with the pointwise topology) if and only if every infinite-dimensional subspace of \(C_p(X)\) contains an isomorphic copy of \(c_{00}\) (with the pointwise topology). Illustrating examples and several open problems are also provided.
