11. Uniform spaces.

The principal notions and propositions relating to uniform spaces came out little by little from the theory of real variables, and were the object of systematic study only recently. Cauchy, looking to set up rigorously the theory of series (cf. p. 153), took as departure point a principle that he seems to have considered as obvious, according to which a necessary and sufficient condition for the convergence of a sequence $(a_n)$ is that $|a_{n+p} - a_n|$ should be arbitrarily small whenever $n$ is sufficiently large (see for example ([56 a], (2), v. VII, p. 267)). With Bolzano [27 c], he was without doubt one of the first to state explicitly this principle, and to recognise its importance: whence the name “Cauchy sequence” given to sequences of real numbers that satisfy the condition in question, and by extension to sequences $(x_n)$ of points in a metric space such that the distance between $x_{n+p}$ and $x_n$ is arbitrarily small whenever $n$ is sufficiently large; from there finally the name of “Cauchy filter” given to the generalisation of Cauchy sequences in uniform spaces.

When later on the intuitive notion of real number was no longer satisfactory, and that it was sought, in order to give Analysis a solid foundation, to define real numbers starting from rational numbers, it was precisely the Cauchy principle that furnished the most fruitful among the definitions suggested in the second half of the XIXth century; it is Cantor’s definition ([47], pp. 93-96) (developed also, among others, from the ideas of Cantor, by Heine [154 b], and independently, by Méray), according to which one makes a real number correspond to every Cauchy sequence (“fundamental sequence” in the terminology of Cantor) of rational numbers; the same real number will correspond to two Cauchy sequences of rational numbers $(a_n)$ and $(b_n)$ if $|a_n - b_n|$ tends to zero, and in this case only. The essential idea here is that, from a certain point of view, the set $\mathbf{Q}$ of rational numbers is “incomplete”, and that the set of real numbers is the “complete” set that is deduced from $\mathbf{Q}$ by “completing” it.

On the other hand, Heine, in work mainly inspired by the ideas of Weierstrass and Cantor, was the first to define uniform continuity for numerical functions of one or more real variables [154 a], and proved that every numerical function, continuous on a closed bounded interval of $\mathbf{R}$, is uniformly continuous there [154 b]: it is “Heine’s theorem”. This result is linked to the compactness of a closed bounded interval in $\mathbf{R}$ (“Borel-Lebesgue theorem", cf. p. 141), and the proof given by Heine of his theorem can also be used, with a few modifications, to prove the Borel-Lebesgue theorem (which seemed to several authors a sufficient reason for giving this theorem the name "Heine-Borel theorem").

The extension of these ideas to more general spaces was made when, first some special cases were studied, and then in general, metric spaces, where a distance (numerical function of pairs of points, satisfying certain axioms) is given and defines at the same time a topology and a uniform structure. Fréchet, who was the first to state the general definition of these spaces, recognised the importance of the Cauchy principle [115 a], and introduced also for all metric spaces, the notion of pre compact (or "totally bounded" [115 a and b]) space. Hausdorff, who, in his "Mengenlehre" [152 a and b] developed a lot of the theory of metric spaces, recognised in particular that one can apply to these spaces the Cantor construction which was considered above, and so deduce, for every non "complete" metric space (that is to say where the Cauchy principle is not valid), a "complete" metric space.

Metric spaces are "uniform spaces" of a particular kind; uniform spaces have only been defined in a general way recently, by A. Weil [330 b]. Beforehand it was only known how to use the notions and results referring to "uniform structure" when metric spaces were being dealt with: which explains the important role played in a lot of modern work on topology by metric or metrizable spaces (and in particular by compact metrizable spaces) in questions where distance is not of any real use. Once the definition of uniform spaces is stated, there is no difficulty (especially when the notion of filter is also available) in extending to these spaces nearly all the theory of metric spaces, as it is stated for example by Hausdorff (and to extend in the same way, for example, to all compact spaces, the results set out for compact metric spaces in the Topology of Alexandroff-Hopf [4]). In particular, the completion theorem for uniform spaces is none other than the transposition, without any essential modification, of Cantor's construction for the real numbers.