BELMANDT is the collective pseudonym under which a group of mathematicians and computer scientists have decide to publish their common works on Pretopology and to give a nod to BOURBAKI. These works would never have been realized without the dynamic leadership of professor Marcel Brissaud, now retired, and who is of course associated to the publication of this book. Marcel Brissaud, during the seventies, has been the first to observe the interest of developing mathematical concepts adapted to humanities. Pretopology theory relies on the statement that the axiomatic of mathematical topology is too strong to model the phenomena of social sciences. This axiomatic has to be reconsidered to give birth to a theory that is adapted to data issued from these fields of expertise, better than trying to force data to fit to an existing and inadequate theory. This book is the first English revised version of a first book, published in French, which is planned to be the first of a series dedicated to the whole set of results obtained in Pretopology: continuity, connectedness, associated metric... Many applications of Pretopology were conducted in various fields: economy with the structural analysis of a economy, game theory with the study of the dynamics of coalitions in cooperative games, sociology with analysis of social networks, data analysis with the construction of clustering algorithms, pattern recognition with digital image analysis as in mathematical morphology, etc. Purely mathematical work in Pretopology has enabled the building of processing algorithms that have been put in a software library, PRETOPOLIB, allowing researchers to implement a pretopological approach in their work. We wish the reader of this book the pleasure of exploring the basics of Pretopology and we hope that he will find the materials for his own work and we give him a rendezvous for the next books on the subject.