Ascoli theorems for Cartesian closed categories. (English) Zbl 0555.18004
Categorical topology, Proc. int. Conf., Toledo/Ohio 1983, Sigma Ser. Pure Math. 5, 599-617 (1984).
[For the entire collection see Zbl 0543.00008.]
The paper contains Ascoli theorems for the (uniform) convergence structure of (uniformly) continuous convergence on function spaces. The Ascoli theorem for uniform convergence spaces makes use of the notion of Ascoli space, introduced in the paper under review. Ascoli spaces and their maps form a bireflective full subcategory ASC of the category PSU of pseudouniform spaces. The author raises several problems concerning this category, e.g. to find an internal characterization of Ascoli spaces and to construct the reflector PSU\(\to ASC\).
The paper contains Ascoli theorems for the (uniform) convergence structure of (uniformly) continuous convergence on function spaces. The Ascoli theorem for uniform convergence spaces makes use of the notion of Ascoli space, introduced in the paper under review. Ascoli spaces and their maps form a bireflective full subcategory ASC of the category PSU of pseudouniform spaces. The author raises several problems concerning this category, e.g. to find an internal characterization of Ascoli spaces and to construct the reflector PSU\(\to ASC\).
Reviewer: C.Némethi
MSC:
| 18B30 | Categories of topological spaces and continuous mappings (MSC2010) |
| 54B30 | Categorical methods in general topology |
| 18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |
| 54C35 | Function spaces in general topology |
