We show that the operator which associates the upper topologyto a pre-order relation can be extended to the morphisms of a subcategory of Top, in such way that it results a functor and it is right adjoint of the respective extension of the operator which associates to each topology its specialization pre-order.

Tipo de documento: Artículo - Article

Palabras clave: Specialization order, Alexandrov topology, upper topology, adjoint functor, adjoint function.

Source: http://www.bdigital.unal.edu.co

## Teaser

Boletı́n de Matemáticas Nueva Serie, Volumen XIII No.
1 (2006), pp.
(∗) Abstract.
We show that the operator which associates the upper topology to a pre-order relation can be extended to the morphisms of a subcategory of Top, in such way that it results a functor and it is right adjoint of the respective extension of the operator which associates to each topology its specialization pre-order. Key words and phrases.
Se muestra que el operador que asocia la topologı́a superior a una relación de pre-orden puede extenderse a los morfismos de una subcategorı́a de Top, de tal manera que resulta un funtor y es adjunto a derecha de la respectiva extensión del operador que asocia a cada topologı́a su pre-orden de especialización. Palabras claves.
Orden de especialización, topologı́a de Alexandrov, topologı́a superior, funtor adjunto, función adjunta. 2000 MSC: Primary 06A15, 54F05, 18A40; Secondary 54A10, 54B30. Introduction Let X be an ordered set.
We say that a topology on X is (order-) concordant if the specialization order (which is defined by x ≤ y ⇔ x ∈ cl{y}) is the given order relation.
Alexandrov studied maximal concordant topologies and called them “discretes” (see [4]).
Later, these topologies, which are characterized like such topologies that are closed under arbitrary intersections, were called Alexandrov topologies or quasi-discrete topologies.
For a given order relation (∗) Lorenzo Acosta G.
Departamento de Matemáticas, Universidad Nacional de Colombia. E-mail: lmacostag@unal.edu.co. 66 T-ADJUNCTION 67 there is also a minimal concordant topology called the upper topology, the upper interval topology or the weak topology.
Concordant topologies are then those topologies which are between the upper topology and the Alexandrov topology of the ordered set.
These topologies are st...