Sets

As stated in the Preface, the aim of this eBook is to study several ordertheoretical tools which are currently used in various fields of mathematics, particularly in set theory, which, in its turn, provides a tool for the whole mathematics. Thus we begin quite naturally with the construction of the essentials of the set-theoretical framework, with special emphasis on order-theoretical aspects.

The first chapter could be more precisely titled “Set theory before the introduction of the concept of order” and comprises three sections. The first one sketches an axiomatic construction of set theory. This is motivated by our belief that although most mathematicians work within the framework of naive set theory, no mathematician could ignore the existence of foundations. However, as the axiomatic line will not be followed in the sequel, the presentation is quite informal. The next section gathers in a systematic presentation the most frequently used properties of correspondences, relations and functions; thus e.g. each of the concepts of injection, surjection and bijection is characterized by 6-8 equivalent conditions. The last section sketches a few categorical prerequisites which will enable us to use the language of categories whenever it will be convenient in the subsequent sections.....