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Grothendieck's theorem on the precompactness of subsets functional spaces over pseudocompact spaces
- Publication Year :
- 2024
-
Abstract
- Generalizations of the theorems of Eberlein and Grothendieck on the precompactness of subsets of function spaces are considered: if $X$ is a countably compact space and $C_p(X)$ is a space of continuous functions in the pointwise topology convergence, then any countably compact subspace of the space $C_p(X)$ is precompact, that is, it has a compact closure. The paper provides an overview of the results on this topic. It is proved that if a pseudo-compact $X$ contains a dense Lindelof $\Sigma$-space, then pseudocompact subspaces of the space $C_p(X)$ are precompact. If $X$ is the product Cech complete spaces, then bounded subsets of the space $C_p(X)$ are precompact. Results on the continuity of separately continuous functions were also obtained.<br />Comment: in russian
- Subjects :
- Mathematics - General Topology
Mathematics - Functional Analysis
Subjects
Details
- Language :
- Russian
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2401.11292
- Document Type :
- Working Paper