A characterization of scattered compact (and $$\omega $$-bounded) spaces

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 $$\tau _{p}$$ τ p and the compact-open topology $$\tau _{k}$$ τ k , respectively. If X is a pseudocompact space, then the...

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Veröffentlicht in:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Jg. 119; H. 4; S. 94
Hauptverfasser: Ka̧kol, Jerzy, López-Alfonso, Salvador, Śliwa, Wiesław
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Milan Springer Nature B.V 01.10.2025
Schlagworte:
Banach spaces
Mathematical functions
Subspaces
Topology
ISSN:1578-7303, 1579-1505
Online-Zugang:Volltext
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Zusammenfassung: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 $$\tau _{p}$$ τ p and the compact-open topology $$\tau _{k}$$ τ k , respectively. If X is a pseudocompact space, then the uniform topology $$\tau _{\infty }$$ τ ∞ on C ( X ) is generated by the sup norm $$\Vert \cdot \Vert _{\infty }$$ ‖ · ‖ ∞ ; clearly $$C_{\infty }(X)=(C(X), \tau _{\infty })=(C(X), \Vert \cdot \Vert _{\infty })$$ C ∞ ( X ) = ( C ( X ) , τ ∞ ) = ( C ( X ) , ‖ · ‖ ∞ ) is a Banach space. We characterize $$\omega $$ ω -bounded (hence also compact) scattered spaces in terms of $$C_{\infty }(X)$$ 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 $$\tau _{\infty }$$ 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.
Bibliographie:ObjectType-Article-1
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ObjectType-Feature-2
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ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-025-01745-w