Abstract
This is a study of the completeness properties of the space C rc ( X) of continuous real-valued functions on a Tychonov space X , where the function space has the C -compact-open topology. Investigate the properties such as completely metrizable, C ech-complete, pseudocomplete and almost C ech-complete.
Highlights
The set-open topology on a family λ of nonempty subsets of the Tychonoff space X is a generalization of the compact-open topology and of the topology of pointwise convergence
Note that the set-open topology and its properties depend on the family λ
We study various kinds of completeness of the C-compact topology such as complete metrizability, Cech-completeness, pseudocompleteness and almost Cech-completeness of Crc(X)
Summary
The set-open topology on a family λ of nonempty subsets of the Tychonoff space X (the λopen topology) is a generalization of the compact-open topology and of the topology of pointwise convergence. The space C(X), equipped with the set-open topology on the family of all C-compact subsets of X, is denoted by Crc(X). The importance of studying the C-compact-open topology on C(X), due to the fact that if Cλ(X) is a locally convex TVS the family λ consists of C-compact subsets of X. A subset O of a space X is called functionally open (or a cozeroset) if X \ O is a zero-set.
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