Exponential objects and Cartesian closedness in the construct \textit{Prtop}
type: Item
instance of: scholarly article
title: Exponential objects and Cartesian closedness in the construct \textit{Prtop}
zbMATH Open document ID: 0796.54019
DOI: 10.1007/BF00872940
published in: Applied Categorical Structures
publication date: 1994-09-28T00:00:00Z
review text: An object \(X\) in a category \({\mathcal K}\) with finite products is called exponential if the functor \(X\times -: {\mathcal K}\to {\mathcal K}\) has a right adjoint. A pretopological space \(X\) is finitely generated if any point \(x\in X\) has a smallest neighborhood. A pretopological space \(X\) is exponential if and only if \(X\) is finitely generated. The full subconstruct of all finitely generated pretopological spaces is a bireflective Cartesian closed subconstruct of the construct Prtop of pretopological spaces. On the other hand, there exists no Cartesian closed coreflective full subconstruct of Prtop containing the space \(\{{1\over n}\); \(n>0\) is an integer\(\}\cup\{0\}\) with the natural topology. Thus the construct FrPrtop whose objects are pretopological spaces such that its pretopology is determined by convergent sequences is not Cartesian closed. A strong Fréchet axiom is given such that the full subconstruct of FrPrTop containing all objects in FrPrtop and satisfying one is a Cartesian closed bireflective subconstruct of FrPrtop.
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 554357
zbMATH Keywords: Fréchet space
zbMATH Keywords: exponential object
zbMATH Keywords: Cartesian closedness
zbMATH Keywords: reflection
zbMATH Keywords: pretopological space
author: Eva Colebunders
author: G. Sonck
reviewed by: Václav Koubek
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00872940
OpenAlex ID: W2131172744
description: scientific article; zbMATH DE number 554357