16. Metric spaces.

As we said (see p. 146), the notion of metric space was introduced in 1906 by M. Fréchet, and developed a few years later by F. Hausdorff in his "Mengenlehre". It acquired great importance after 1920, on the one hand following the fundamental work of S. Banach and his school on normed spaces and their applications to functional Analysis (see pp. 216 ff.), on the other by reason of the interest that was raised by the notion of valuation in Arithmetic and in Algebraic Geometry (where notably completion with respect to a valuation shows itself to be very fruitful).

From the period 1920-1930 are dated a whole series of studies undertaken by the Moscow school on the properties of the topology of a metric space, work which aimed in particular at obtaining necessary and sufficient conditions for a given topology to be metrizable. It is this movement of ideas that caused the appearance of interest in the notion of a normal space, defined in 1923 by Tietze, but whose important role was only recognised following the work of Urysohn [314] on the extension of continuous numerical functions. Apart from the trivial case of functions of a real variable, the problem of extension to the whole space of a continuous numerical function defined on a closed set, was treated for the first time (for the case of the plane) by H. Lebesgue [196 d]; before the definitive result of Urysohn, it had been solved for metric spaces by H. Tietze [307]. The extension of this problem to the case of functions with values in an arbitrary topological space acquired in these last years a considerable importance in algebraic Topology. Recent works have as well brought into the open that, in this type of question, the notion of normal space is hard to handle, because it still offers too many "pathological" possibilities; most of the time it is necessary to substitute for it the more restrictive notion of a paracompact space, introduced by J. Dieudonné in 1944 [90 a]; in this theory, the most remarkable result is the theorem due to A. H. Stone [300], according to which every metrizable space is paracompact.$^1$

$^1$ This theorem has allowed the provision of a more satisfying solution to the problem of metrization than the criteria obtained around 1930 by the Russo-Polish school ("Nagata-Smirnov criterion"). But it must be noted that until now these criteria have hardly had any applications; as so often in the history of mathematics it seems as if the problem of metrization had less importance because of its solution than for the new notions whose development it would have brought.

We have already focused attention (see p. 155) on the important works of the end of the XIXth century and of the beginning of the XXth century (E. Borel, Baire, Lebesgue, Osgood, W. H. Young) on the classification of sets of points in the spaces $\mathbf{R}^n$, and on the classification and characterisation of numerical functions obtained by iterating, starting with continuous functions, the process of passing to the limit (for sequences of functions). It rapidly became apparent that metric spaces supplied a natural framework for research of this kind, whose development after 1910 is mainly due to the Russian and Polish schools. It is these schools that have between them brought to light the fundamental role played in modern Analysis by the notion of thin sets, and by the theorem on the countable intersection of open sets everywhere dense in a complete metric space, proved first (independently) by Osgood [240] for the numerical straight line and by Baire [11 b] for the spaces $\mathbf{R}^n$.

On the other hand, in 1917, Souslin [289], correcting an error of Lebesgue, showed that the continuous image of a Borel set is not necessarily Borel, which led him to the definition and study of the larger category of sets, since called "analytic" or "Souslin"; after the premature death of Souslin this study was mainly undertaken by N. Lusin (whose ideas had inspired the work of Souslin) and by Polish mathematicians (see [209] and [189 b]). The actual importance of these sets relies mainly on their applications in the theory of integration (where, thanks to their special properties, they allow constructions that would be impossible on arbitrary measurable sets), and in modern potential theory, where the fundamental theorem on the capacity of Souslin sets, proved recently by G. Choquet [63], has already shown itself to be rich in varied applications.