On relations between X and Cc(X)

Feldman, Wilhelm

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URL: https://ub-madoc.bib.uni-mannheim.de/1920
URN: urn:nbn:de:bsz:180-madoc-19202
Document Type: Working paper
Year of publication: 1971
Place of publication: Mannheim
Publication language: English
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Mathematik und Informatik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Mathematik > Mannheimer Manuskripte
Subject: 510 Mathematics
Subject headings (SWD): Relation <Mathematik>
Abstract: The algebra of all continuous real-valued functions on a space X endowed with the continuous convergence structure is denoted by Cc(x) . Relationships between a space X and its associated convergence algebra Cc(X) are investigated. After appropriate definitions, the following two theorems are proved: (1). A c-embedded convergence space X is Lindelöf if and only if Cc(X) is first countable (this has a generalization to upper &#935;-compact spaces). (2). A c-embedded convergence space X has weight &#935; if and only if Cc(X) has weight &#935;. With the help of (2), it is shown that a completely regular topological space X is separable and metrizable if and only if Cc(X) is second countable. A type of Stone-Weierstrass theorem proved by E. Binz is extended to deal with questions of density. This extension is utilized to provice another characterization of separable metrizable spaces, and to show that the algebraic tensor product of C(X) and C(Y) may be regarded as a dense subalgebra of Cc(X x Y). An inductive limit (in the category of convergence spaces) of certain locally convex topological vector spaces is constructed. This inductive limit proves to be a useful approximation of Cc(X). However, for a wide class of topological spaces, it is shown that Cc(X) can not even be realized as an inductive limit of topolagical vector spaces.
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