Bourbaki-complete spaces and Samuel realcompactification

Abstract

This thesis belongs to the area of General Topology and, in particular, to the field of study of uniform spaces. It is divided in three parts where the related topics Bourbaki-completeness and Samuel realcompactification are studied. Many of the results presented here have already been published by the author in the following papers. [GaMe14] M. I. Garrido and A. S. Mero~no, New types of completeness in metric spaces, Ann. Acad. Sci. Fenn. Math. 39 (2014) 733-758. [GaMe16] M. I. Garrido and A. S. Mero~no, On paracompactness, completeness and boundedness in uniform spaces, Topology Appl. 203 (2016)98-107. [GaMe17] M. I. Garrido and A. S. Mero~no, The Samuel realcompactification of a metric space, J. Math. Anal. Appl. 456 (2017) 1013-1039. [GaMe18] M. I. Garrido and A. S. Mero~no, The Samuel realcompactification,Topology Appl. 241 (2018) 150-161. [HoJuMe19] A. Hohti, H. Junnila and A. S Mero~no, On strongly Cechcomplete spaces, to appear in Topology Appl. (2019). In addition, new results, originated during the writing process, have been included. In the first part of the thesis we present many results of [GaMe14], [GaMe16] and some of [HoJuMe19]. More precisely, we study Bourbaki-completeness and cofinal Bourbaki-completeness (equivalently uniform strong-paracompactness) of uniform spaces. In particular, we solve several primary problems related to products, subspaces, hyperspaces, metric spaces and fine spaces...Esta tesis pertenece al area de la Topología General y en particular, al campo de estudio de los espacios uniformes. Esta dividida en tres partes donde se estudian los temas relacionados de Bourbaki-completitud y Realcompactificacion de Samuel. Muchos de los resultados presentados aquí aparecen en las siguientes publicaciones del autor. [GaMe14] M. I. Garrido and A. S. Mero~no, New types of completeness in metric spaces, Ann. Acad. Sci. Fenn. Math. 39 (2014) 733-758. [GaMe16] M. I. Garrido and A. S. Mero~no, On paracompactness, completeness and boundedness in uniform spaces, Topology Appl. 203 (2016)98-107. [GaMe17] M. I. Garrido and A. S. Mero~no, The Samuel realcompactification of a metric space, J. Math. Anal. Appl. 456 (2017) 1013-1039. [GaMe18] M. I. Garrido and A. S. Mero~no, The Samuel realcompactification,Topology Appl. 241 (2018) 150-161. [HoJuMe19] A. Hohti, H. Junnila and A. S Mero~no, On strongly Cech-complete spaces, to appear in Topology Appl. (2019). Tambien han sido incluidos resultados nuevos, surgidos durante el proceso de escritura de la tesis. En la primera parte de esta memoria aparecen muchos resultados de los trabajos [GaMe14], [GaMe16] y algunos de [HoJuMe19]. Mas precisamente, estudiamos la Bourbaki-completitud y la cofinal Bourbaki-completitud (equivalentemente, la fuerte paracompacidad uniforme) en los espacios uniformes. En particular, resolvemos varios problemas basicos relacionados con productos, subespacios, hiperespacios, espacios metricos y espacios con la uniformidad fina...