On démontre le théorème suivant: pour qu'un espace topologique Y quelconque, même pas séparé, ait ses ouverts localement compacts, il faut et il suffit que le foncteur "produit par Y" soit canoniquement adjoint au foncteur "Hom (Y , )" construit par la topologie compacte-ouverte. Le théorème est généralisé aux espaces fibrés sans structure.
Let be a set, a topology on is completely regular if, and only if, is the topology defined by a family of maps . It is not difficult to prove that in some sense is minimal under this condition. The purpose of this paper is to characterize the spaces of values that are minimal and the families of topologies on that are complete under the property of being induced by a family of maps .
A notion of hereditarity of a closure operator with respect to a class of monomorphisms is introduced. Let be a regular closure operator induced by a subcategory . It is shown that, if every object of is a subobject of an -object which is injective with respect to a given class of monomorphisms, then the closure operator is hereditary with respect to that class of monomorphisms.
We introduce the concept of firm classes of morphisms as basis for the axiomatic study of completions of objects in arbitrary categories. Results on objects injective with respect to given morphism classes are included. In a finitely well-complete category, firm classes are precisely the coessential first factors of morphism factorization structures.
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