New types of completeness in metric spaces

This paper is devoted to introduce and study two new properties of completeness in the setting of metric spaces. We will call them Bourbaki-completeness and cofinal Bourbaki-completeness. These notions came from new classes of generalized Cauchy sequences appearing when we try to characterize the so-called Bourbaki-boundedness in a similar way that Cauchy sequences characterize the totally boundedness. We also study the topological problem of metrizability by means of a Bourbaki-complete or a cofinally Bourbaki-complete metric. At this respect, we obtain results in the line to the classical Cech theorem about the complete metrizability of a metric space X in terms of its Stone-Cech compactification beta X. Finally we give some relationships between these kinds of completeness and some properties related to paracompactness and uniform paracompactness in the framework of metrizable spaces