Abstract
What is known about exponentials in categories of pretopological spacesdepends essentially on calculations and constructions with non-Hausdorffspaces, especially the space 3 consisting of 3 points, only one of whichadmits a nontrivial (pre-)neighborhood filter with exactly 2 neighborhoods.Exponentials in coreflective subcategories C of Prtop, the category of all pretopological spaces, are known to be always finitely generated, andvice versa, if C is finitely productivein Prtop. This result can be reproved using a completely different and veryelementary approach which works for Hausdorff spaces as well as for otherseparation properties and generalizes to regular epireflectivesubcategories. Moreover, every finitely productive topological subconstructof Prtop, that contains some not finitely generated net space, turns out tobe not Cartesian closed. This extends the respective result for the specialnet space, N¯, the convergent sequence.
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Richter, G. More on Exponential Objects in Categories of Pretopological Spaces. Applied Categorical Structures 5, 309–319 (1997). https://doi.org/10.1023/A:1008637703914
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DOI: https://doi.org/10.1023/A:1008637703914