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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| Published in: | Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 119; no. 4; p. 94 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Milan
Springer Nature B.V
01.10.2025
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| Subjects: | |
| ISSN: | 1578-7303, 1579-1505 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1578-7303 1579-1505 |
| DOI: | 10.1007/s13398-025-01745-w |