Introduction

Most branches of mathematics involve structures of a type different from the algebraic structures (groups, rings, fields, etc.) which are the subject of the book Algebra of this series: namely structures which give a mathematical content to the intuitive notions of limit, continuity and neighbourhood. These structures are the subject matter of the present book.

Historically, the ideas of limit and continuity appeared very early in mathematics, notably in geometry, and their role has steadily increased with the development of analysis and its applications to the experimental sciences, since these ideas are closely related to those of experimental determination and approximation. But since most experimental determinations are measurements, that is to say determinations of one or more numbers, it is hardly surprising that the notions of limit and continuity in mathematics were featured at first only in the theory of real numbers and its outgrowths and fields of application (complex numbers, real or complex functions of real or complex variables, Euclidean geometry and related geometries).

In recent times it has been realized that the domain of applicability of these ideas far exceeds the real and complex numbers of classical analysis (see the Historical Note to Chapter I). Their essential content has been extracted by an effort of analysis and abstraction, and the result is a tool whose usefulness has become apparent in many branches of mathematics.

In order to bring out what is essential in the ideas of limit, continuity and neighbourhood, we shall begin by analysing the notion of neighbourhood (although historically it appeared later than the other two). If we start from the physical concept of approximation, it is natural to say that a subset $A$ of a set $E$ is a neighbourhood of an element $a$ of $A$ if, whenever we replace $a$ by an element that "approximates" $a$, this new element will also belong to $A$, provided of course that the "error" involved is small enough; or, in other words, if all the points of E which are "sufficiently near" $a$ belong to A. This definition is meaningful whenever precision can be given to the concept of sufficiently small error or of an element sufficiently near another. In this direction, the first idea was to suppose that the "distance" between two elements can be measured by a (positive) real number. Once the "distance" between any two elements of a set has been defined, it is clear how the "neighbourhoods" of an element $a$ should be defined: a subset will be a neighbourhood of $a$ if it contains all elements whose distance from $a$ is less than some preassigned strictly positive number. Of course, we cannot expect to develop an interesting theory from this definition unless we impose certain conditions or axioms on the "distance" (for example, the inequalities relating the distances between the three vertices of a triangle which hold in Euclidean geometry should continue to hold for our generalized distance). In this way we arrive at a vast generalization of Euclidean geometry. It is convenient to continue to use the language of geometry: thus the elements or a set on which a "distance" has been defined are called points, and the set itself is called a space. We shall study such spaces in Chapter IX.

So far we have not succeeded in freeing ourselves from the real numbers. Nevertheless, the spaces so defined have a great many properties which can be stated without reference to the "distance" which gave rise to them. For example, every subset which contains a neighbourhood of $a$ is again a neighbourhood of $a$, and the intersection of two neighbourhoods of $a$ is a neighbourhood of $a$. These properties and others have a multitude of consequences which can be deduced without any further recourse to the "distance" which originally enabled us to define neighbourhoods. We obtain statements in which there is no mention of magnitude or distance.

We are thus led at last to the general concept of a topological space, which does not depend on any preliminary theory of the real numbers. We shall say that a set E carries a topological structure whenever we have associated with each element of E, by some means or other, a family of subsets of E which are called neighbourhoods of this element — provided of course that these neighbourhoods satisfy certain conditions (the axioms of topological structures). Evidently the choice of axioms to be imposed is to some extent arbitrary, and historically has been the subject of a great deal of experiment (see the Historical Note to Chapter I). The system of axioms finally arrived at is broad enough for the present needs of mathematics, without falling into excessive and pointless generality.

A set carrying a topological structure is called a topological space and its elements are called points. The branch of mathematics which studies topological structures bears the name of Topology (etymologically, "science of place", not a particularly expressive name), which is preferred nowadays to the earlier (and synonymous) name of Analysis situs.

To formulate the idea of neighbourhood we started from the vague concept of an element "sufficiently near" another element. Conversely, a topological structure now enables us to give precise meaning to the phrase "such and such a property holds for all points sufficiently near $a$" : by definition this means that the set of points which have this property is a neighbourhood of $a$ for the topological structure in question.

From the notion of neighbourhood there flows a series of other notions whose study is proper to topology : the interior of a set, the closure of a set, the frontier of a set, open sets, closed sets, and so on (see Chapter I, § 1). For example, a subset $A$ is an open set if, whenever a point $a$ belongs to $A$, all the points sufficiently near $a$ belong to $A$; in other words, if $A$ is a neighbourhood of each of its points. The axioms for neighbourhoods have certain consequences for all these notions; for example, the intersection of two open sets is an open set (because we have supposed that the intersection of two neighbourhoods of $a$ is a neighbourhood of $a$). Conversely, we can start from one of these derived notions instead of starting from the notion of a neighbourhood; for example, we may suppose that the open sets are known, and take as axioms the properties of the family of open sets (one of these properties has just been stated, by way of example). We can then verify that, from knowledge of the open sets, the neighbourhoods can be reconstructed; the axioms for neighbourhoods are now consequences of the new axioms for open sets that we took as a starting point. Thus a topological structure can be defined in various different ways which are basically equivalent. In this book we shall start from the notion of open set, because the corresponding axioms are the simplest.

Once topological structures have been defined, it is easy to make precise the idea of continuity. Intuitively, a function is continuous at a point if its value varies as little as we please whenever the argument remains sufficiently near the point in question. Thus continuity will have an exact meaning whenever the space of arguments and the space of values of the function are topological spaces. The precise definition is given in Chapter I, § 2.

As with continuity, the idea of a limit involves two sets, each endowed with suitable structures, and a mapping of one set into the other. For example, the limit of a sequence of real numbers $a_n$ involves the set $\mathbf{N}$ of natural numbers, the set $\mathbf{R}$ of real numbers, and a mapping of the former set into the latter. A real number $a$ is then said to be a limit of the sequence if, whatever neighbourhood $V$ of $a$ we take, this neighbourhood contains all the $a_n$ except for a finite number of values of $n$; that is, if the set of natural numbers $n$ for which $a_n$ belongs to $V$ is a subset of $\mathbf{N}$ whose complement is finite. Note that $\mathbf{R}$ is assumed to carry a topological structure, since we are speaking of neighbourhoods; as to the set $\mathbf{N}$, we have made a certain family of subsets play a particular part, namely those subsets whose complement is finite. This is a general fact: whenever we speak of limit, we are considering a mapping $f$ of a set $E$ into a topological space $F$, and we say that $f$ has a point $a$ of $F$ as a limit if the set of elements $x$ of $E$ whose image $f(x)$ belongs to a neighbourhood $V$ of $a$ [this set is just the "inverse image" $f^{-1}(V)$] belongs, whatever the neighbourhood $V$, to a certain family $\mathfrak{F}$ of subsets of $E$, given beforehand. For the notion of limit to have the essential properties ordinarily attributed to it, the family $\mathfrak{F}$ must satisfy certain axioms, which are stated in Chapter I, . Such a family $\mathfrak{F}$ of subsets of $E$ is called a filter on $E$. The notion of a filter, which is thus inseparable from that of a limit, appears also in other contexts in topology; for example, the neighbourhoods of a point in a topological space form a filter

The general study of all these notions is the essential purpose of Chapter I. In addition, particular classes of topological spaces are considered there, spaces which satisfy more restrictive axioms, or spaces obtained by particular procedures from other given spaces.

As we have already said, a topological structure on a set enables one to give an exact meaning to the phrase "whenever $x$ is sufficiently near $a$, $x$ has the property $P\{x\}$". But, apart from the situation in which a "distance" has been defined, it is not clear what meaning ought to be given to the phrase "every pair of points $x, y$ which are sufficiently near each other has the property $P\{x, y\}$", since a priori we have no means of comparing the neighbourhoods of two different points. Now the notion of a pair of points near to each other arises frequently in classical analysis (for example, in propositions which involve uniform continuity). It is therefore important that we should be able to give a precise meaning to this notion in full generality, and we are thus led to define structures which are richer than topological structures, namely uniform structures. They are the subject of Chapter II.

The other chapters of this Book are devoted to questions in which, in addition to a topological or uniform structure, there is some other structure present. For example a group which carries a suitable topology (compatible in a certain sense with the group structure) is called a topological group. Topological groups are studied in Chapter III, and we shall see there in particular how every topological group can be endowed with certain uniform structures.

In Chapter IV we apply the preceding principles to the field of rational numbers. This enables us to define the field of real numbers; because of its importance, we study it in considerable detail. In the succeeding chapters, starting from the real numbers, we define certain topological spaces which are of particular interest in applications of topology to classical geometry: finite-dimensional vector spaces, spheres, projective spaces, etc. We consider also certain topological groups closely related to the additive group of real numbers, which we characterize axiomatically, and this leads us to the definition and elementary properties of the most important functions of classical analysis: the exponential, logarithmic and trigonometric functions.

In Chapter IX we revert to general topological spaces, but now with a new instrument, namely the real numbers, at our disposal. In particular we study spaces whose topology is defined by means of a "distance"; these spaces have properties, some of which cannot be extended to more general spaces. In Chapter X we study sets of mappings of a topological space into a uniform space (function spaces); these sets, suitably topologized, have interesting properties which already play an important part in classical analysis.