Sets and Topology
Introduction to part I : Set Theory
The concept of a set is one of the most fundamental and most frequently used mathematical concepts. In every domain of mathematics we have to deal with sets such as the set of positive integers, the set of complex numbers, the set of points on a circle, the set of continuous functions, the set of integrable functions, and so forth.
The object of set theory is to investigate the properties of sets from the most general point of view; generality is an essential aspect of the theory of sets. In geometry we consider sets whose elements are points, in arithmetic we consider sets whose elements are numbers, in the calculus of variations we deal with sets of functions or curves; on the other hand, in the theory of sets we are concerned with the general properties of sets independently of the nature of the elements which comprise these sets. This will made clear by several examples which we shall give here and by a brief overall view of the contents of the first part of this volume.
In Chapter II we shall consider operations on sets which are analogous to arithmetic operations: for every pair of sets A and B we shall form their union A ∪ B, understanding by this the set composed of all elements of the set A and all elements of the set B; we shall also form the intersection A ∩ B of the sets A and B, and we shall understand by this the set of all elements common to the sets A and B. These operations have, in a certain sense, an algebraic character, e. g. they have the properties of commutativity, associativity, and distributivity. It is clear that these properties do not depend on whether these sets consist of numbers, points or other mathematical objects; they are general properties of sets and therefore the investigation of these properties belongs to the realm of set theory.
In Chapter III we consider another type of operation, called cartesian multiplication. For two given sets X and Y we denote by X × Y the set of all pairs of elements (x, y) in which the first belongs to the set X and the second to the set Y. Thus, e. g. if X and Y denote the set of real numbers then X × Y is the plane (whence the name "cartesian product" in honour of the great French. mathematician Descartes (1596-1650), who, treating the plane as a set of pairs of real numbers, initiated a new branch of mathematics, called analytic geometry). The computational properties of cartesian multiplication in connection with the operations on sets mentioned above are given in Chapter III.
The concept of cartesian product allows us to define the concept of a function in a completely general way. We shall concern, ourselves with the concept of function in Chapter IV. An especially important role in the theory of sets is played by the one-to-one functions. These are functions which map the set X onto the set Y so that to every two distinct elements of the set X there correspond two distinct elements of the set Y (and then the inverse function with respect to the given function, which maps the set Y onto the set X, is also one-to-one). If there exists such a one-to-one mapping of the set X onto the set Y we say that these sets are of equal power. The equality of powers is the generalization of the idea of equal number of elements; the significance of this generalization depends first of all on the fact that it can be applied to infinite as well as to finite sets. For example, it is easy to see that the set of all even numbers has the same power as the set of all odd numbers; on the other hand, the set of all real numbers does not have the same power as the set of all natural numbers - a fact which is not immediately obvious. Hence, we can - in some sense - classify infinite sets with respect to their power. We can also, thanks to this, extend the sequence of natural numbers, introducing numbers which characterize the power of infinite sets (called the cardinal numbers); in particular, to sets having the same power as the set of all natural numbers (or the countably infinite sets) we assign the cardinal number a to the set of all real numbers we assign the number c (the power of the continuum). It turns out that there is an infinite number of infinite cardinal numbers. However, in the applications of set theory to other branches of mathematics an essential role is played by only two of them: a and c. So we also limit ourselves above all to the investigation of these two numbers. This forms the content of Chapter V and VI.
Chapter VII is devoted to ordered sets such as the set of all natural numbers, the set of all rational numbers, the set of all real numbers. For each of these sets the "less than" relation determines the ordering; here the order types of these three sets differ in an essential manner: in the first of them there exist elements which are immediately adjacent to one another (n and n +1), in the second there are no such elements (so we say, the ordering is dense), however, there exist gaps (in the Dedekind sense), but in the set of all real numbers there are no gaps.
An especially important kind of ordered sets are the well ordered sets, i. e. those whose every non-empty subset has a least element. An example of a well ordered set is the set of all natural numbers (but the set of all integers is not well ordered since this set does not have a least element). Also well ordered - although of a different order type - is the set consisting of numbers of the form 1 - 1/n and numbers of the form 2 - 1/n, n = 1, 2, 3, ... In Chapter VIII we give the most important theorems concerning well ordering. Among other things, we prove that of two distinct order types of well ordered sets one is always an extension of the other (in a sense which we shall make more precise). From this follows the important corollary that of two different well ordered sets one is of power equal to that of a subset of the other; in the terminology of cardinal numbers this means that for two distinct cardinal numbers corresponding' to well ordered sets, one is always smaller than the other. In connection with this theorem, there arises the fundamental conjecture: does there exist a relation for any set which establishes its well ordering? - We shall prove that this is in fact so, if we assume, the axiom of choice. This theorem is the final theorem of the first part of this book.
The discussion of set theory given here is based on a system of axioms. Even though in the introductory part of set theory, e. g. in the algebra of sets, the concept of set, with which we have to deal in mathematics (and hence the concept of a set of numbers, points or curves, and so on) is such that it does not touch upon logical difficulties, a subsequent construction of set theory which is not based on a system of axioms turns out to be impossible; for there exist questions, to which the so-called "naive" intuitive idea of a set does not give a unique answer. The lack of the necessary foundations of set theory in its initial period of development led to the so-called antinomies, i. e. contradictions, which one did not know how to interpret on the basis of the "naive" intuitive idea of set. Only the axiomatic concept of the theory of sets allowed the removal of these antinomies.
In the present book we do not analyse more closely the axiomatics of set theory or the logical foundation of the subject. Although these subjects form at the present time an important part of mathematics and are being actively developed, the discussion of them in this book lies outside the principal goal of the book which is: the presentation of the most important branches of set theory and topology from the point of view of their applications to other branches of mathematics.
In the first part of this book the reader will find a certain amount of information on mathematical logic. The notation of mathematical logic is an indispensable tool of set theory and can be applied with great profit far beyond set theory. In Chapters I and III we have given the main facts from this subject concerning the calculus of propositions, propositional functions and quantifiers. The notation of mathematical logic is not devoid of general didactical values; by examples for concepts such as uniform convergence or uniform continuity it is possible to observe how much the definition of these concepts gains in precision and lucidity, when they are written in the symbolism of mathematical logic.
In the first period of its existence, set theory was practically exclusively the creation of one scholar, G Cantor (1845-1918). In the period preceding the appearance of the works of Cantor, there were published works containing concepts which are now included in the theory of sets (by authors such as Dedekind, Du Bois-Reymond, Bolzano), but nonetheless the systematic investigation of the general properties of sets, the establishment of fundamental definitions and theorems and the creation on their foundation of a new mathematical discipline is the work of G Cantor (during the years 1871-1883).
The stimulus to the investigations from which the theory of sets grew, was given by problems of analysis, the establishing of the foundations of the theory of irrational numbers, the theory of trigonometric series, etc, However, the further development of set theory went initially in an abstract direction, little connected with other branches of mathematics. This fact, together with a certain strangeness of the methods of set theory which were entirely different from those applied up to that time, caused many mathematicians to regard this new branch of mathematics initially with a certain degree of distrust and reluctance. In the course of years, however, when set theory showed its usefulness in many branches of mathematics such as the theory of analytic functions or theory of measure, and when it became an indispensable basis for new mathematical disciplines (such as topology, the theory of functions of a real variable, the foundations of mathematics), it became an especially important branch and tool of modern mathematics.