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ELEMENTS OF POINT-SET TOPOLOGY

Carlos Prieto

2010

Date of version: October 28, 2009

Preface

Nowadays one speaksmore and more about the specialization in modern sci- ence. Even though this statement is valid up to a certain point, one might say that a characteristic of science today is the every time greater interaction among the various disciplines that conform it. Similarly to to what happens in science in general, in each discipline one pursues a broader relationship among the di®erent ¯elds that conform it. In mathematics, for instance, one expects from a di®erential geometer or from a function theorist a much wider common knowledge than the one required one half century ago. This happens because of the ubiquity that some mathematical concepts show more and more. One of these mathematical concepts is that of a topological space, that includes everything related with \nearness", \continuity", \neighborhood", \deformation", et cetera. Topology has been for many years one of the most important and in°uential ¯elds in modern mathematics. Its origins date back over some centuries, but it was out the twentieth century. There are other great names among those who created point-set topology, whose existence has been justi¯ed by the great progress of alge- braic topology. On the one hand, the e®ectiveness of point-set topology, more than due to deep theorems, it rests in the ¯rst place on its conceptual simplicity and on its convenient terminology, because in a sense it establishes a link between abstract, not very intuitive problems, and our ability to visualize geometric phenomena in space. This intellectual ability to grasp what is going on in 3-dimensional space, that through topology allows us to delve into mathematical thinking and into the world of abstract objects, is very independent of abstraction and logical thinking. This reinforcement of our mathematical talent is probably the deepest cause of the e®ectiveness and the simplicity of the topological methods. As many of the basic mathematical branches, topology has an intricate his- tory. If we mark the start of topology at the point when the conceptual system of point-set topology was established, then we have to refer to Felix Hausdor®'s bookGrundzÄuge der Mengenlehre(Foundations of Set Theory), Leipzig, 1914, in whose Chapter 7 \Point Sets in General Spaces", he establishes the most impor- tant and basic concepts in point-set topology. Already in 1906, in his paperSur ivPreface quelques points du calcul fonctionnel(On some Topics of Functional Calculus), the concept of topological space, by giving an axiomatic approach to the concept functional analysis. But, of course, the history dates further back to the times when the e®er- vescence of geometry was taking place during the nineteenth century. At the beginning of that century there was the classical idea that geometry was the math- ematical ambit, where the concepts of the physical space developed. Towards the end of that century, as it was shown by Felix Klein in hisErlanger Programm (Erlangen Program. Comparative Considerations about New Geometric Investi- gations), the projection went much farther than the physical space and it even started to consider such abstract spaces as then-manifolds, the projective spaces, the Riemann surfaces, or even the function spaces. Among the decisive works for the emergence of topology, one ¯nds the mon- umental work of Georg Cantor. He established the bases on which the abstract concept of a topological space is formulated as \a set furnished with a collection of subsets such that..." Indeed, already in 1870, Cantor had shown that if two Fourier series converge pointwise and have the same limit, then they must have the same coe±cients. Cantor himself improved this result in 1871 by showing that the coincidence of the coe±cients can equally be achieved by requiring pointwise convergence or equality of the limits, up to a ¯nite set in the interval [0;2¼]. In

1872 he analyzed certain in¯nite subsets, up to which his statement remains valid.

It was then, when he introduced his famousCantor set, that being \only" a subset of an interval, it is topologically not only a very interesting object, but of great importance in several branches of mathematics. The problem of deciding if two spaces are homeomorphic or not is no doubt the central problem in topology. It was not until the creation of algebraic topology that it was possible to give a reasonable answer to such a problem. Now it is not only because of its conceptual simplicity and its adequate symbology, but thanks to the powerful tool provided by algebra and its most convenient functorial relationship to topology that this e®ectiveness is achieved. The analytic description of dynamical systems in classical mechanics repre- sented the ¯rst step towards the necessity to create a geometrical language in di- mensions higher than the usual ones. Already in the eighteenth century, Lagrange, the possibility of grasping a fourth dimension. It was Riemann, in his famousHa- bilitationsvortrag: ÄUber die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypotheses underlying Geometry), GÄottingen, 1854, who presented the

Prefacev

¯rst ideas on the geometry of manifolds.

Lagrange himself, in hisLe»cons sur le Calcul des Fonctions(Lessons on Func- tional Calculus), Paris, 1806, introduced the concept of perturbation, or homotopy, of curves in variational calculus problems to detect certain minimal curves. What

1899, London 1900, Paris 1902, Paris 1902, and Palermo 1904. In the ¯rst, he

notices that \geometry is the art of reasoning well with badly made ¯gures." And further says: \Yes, without doubt, but under one condition. The proportions of the ¯gures might be grossly altered, but their elements must not be interchanged and must preserve their relative situation. In other terms, one does not worry about quantitative properties, but one must respect the qualitative properties. That is to say precisely those, which are the concern of Analysis Situs." Indeed, other This is the case for his memoir on the qualitative theory of di®erential equations, in topological terms the famous Euler formula, and constitutes one of the ¯rst maps on manifolds such as, for instance, vector ¯elds, whose indexes determine the question on the classi¯cation of manifolds having in mind the classi¯cation of the orientable surfaces considered by Moebius in hisTheorie der elementaren Verwandtschaft(Theory of elementary relationship), Leipzig, 1863. This classi¯- the deformation of surfaces), Paris, 1866, who, by classifying surfaces solved an important homeomorphism problem. Jordan also studied the homotopy classes of closed paths, that is, the ¯rst notions of the fundamental group, inspired by Riemann, who already had ana- lyzed the behavior of integrals of holomorphic di®erential forms and therewith the concept of homological equivalence between closed paths. history is in itself the object of another text. Doubtless, the text [10] edited by I.M. James is an excellent reference in that direction. This book has the purpose of presenting the topics of point-set topology, which from my own point of view, are basic for an undergraduate student, who is inter- ested in this area or a±ne areas in mathematics. The design of the text is as follows. We start with a small rather motivating Chapter 1, followed by six substantial chapters, each of which is divided into viPreface several sections that are distinguished by their double numbering (1.1, 1.2, 2.1, :::). De¯nitions, propositions, theorems, remarks, formulas, exercises, etc., are designated with triple numbering (1.1.1, 1.1.2,:::). Exercises are an important part of the text, since many of them are intended to carry the reader further along the lines already developed, in order to prove results that are either important by themselves or relevant for future topics. Most of these are numbered, but occasionally they are identi¯ed inside the text by the use of italics (exercise). The book starts considering metric spaces to arrive to the abstract properties of their open sets. They lead us to the abstract concept of a topological space.quotesdbs_dbs7.pdfusesText_5

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