A characterization of scattered compact (and $$ω $$-bounded) spaces - Ka̧kol, Jerzy; López-Alfonso, Salvador; Śliwa, Wiesław (Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas, 2025)
For a Tychonoff space X by $$C_p(X)$$ C p ( X ) and $$C_k(X)$$ C k ( X ) we denote the space C ( X ) of continuous real valued functions on X endowed with the pointwise topology $$τ _{p}$$ τ p and the compact-open topology $$τ _{k}$$ τ k , respectively. If X is a pseudocompact space, then the uniform topology $$τ _{∞ }$$ τ ∞ on C ( X ) is generated by the sup norm $$\Vert · \Vert _{∞ }$$ ‖ · ‖ ∞ ; clearly $$C_{∞ }(X)=(C(X), τ _{∞ })=(C(X), \Vert · \Vert _{∞ })$$ C ∞ ( X ) = ( C ( X ) , τ ∞ ) = ( C ( X ) , ‖ · ‖ ∞ ) is a Banach space. We characterize $$ω $$ ω -bounded (hence also compact) scattered spaces in terms of $$C_{∞ }(X)$$ C ∞ ( X ) and $$C_p(X)$$ 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 $$C_p$$ 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 $$τ _{∞ }$$ τ ∞ -closed subspace of $$C_p(X)$$ C p ( X ) contains an isomorphic copy of $$c_0$$ c 0 (with the pointwise topology) if and only if every infinite-dimensional subspace of $$C_p(X)$$ C p ( X ) contains an isomorphic copy of $$c_{00}$$ c 00 (with the pointwise topology). Illustrating examples and several open problems are also provided.