392 12. MODERN GEOMETRIESincluding Henri Lebesgue (1875 1941), and is now generally known as the Heine-
Borel theorem. The word compact was first used in 1906 in two equivalent senses
in two different papers, by Maurice Frechet (1878-1973).
Closed and open sets. The word set, without which modern mathematicians would
not be able to talk at all, was not introduced formally until the 1870s. The history
of set theory is discussed in more detail in Chapter 19. At present we merely
mention that the idea of a closed set arose from consideration of the set of limit
points of a given set (its derived set). In an 1884 paper, Georg Cantor (1854-1918)
called a set closed if it contained all of its limit points. Since it was easy to show
that a limit point of limit points of a set Ñ is itself a limit point of P, it followed
that the derived set P' is always a closed set.
Although the phrase closed set appears in 1884, its dual—the phrase open set—
did not appear for nearly two more decades. Weierstrass had used the concept of an
open set in discussing analytic functions, since he used power series, which required
that the function be defined a small disk called a neighborhood about each point
in its domain of definition. Weierstrass used the German term Gebiet (region) for
such a domain of definition. The phrase open set seems to have been used for the
first time by W. H. Young in 1902.^48 In a 1905 paper in descriptive function theory
(that is, discussing what it means for a function to be "analytic" in a very general
sense), Henri Lebesgue referred specifically to ensembles ouverts (open sets) and
defined them to be the complements of closed sets.
Metric spaces. The frequent repetition of certain basic patterns of reasoning, and
perhaps just a normal human penchant for order, led to the creation of very abstract
structures around the beginning of the twentieth century. The kind of continuity
argument we now associate with <5's and e's was generalized in 1905 by Maurice
Frechet, who considered abstract sets on which there was a sort of distance between
two points A and B, denoted (A, B). This distance had the properties normally
associated with distance, that is, (Á, Â) = (B, A) (the distance from A to  is the
same as the distance from  to A), (A, B) > 0 if Á ö Â, and (A, A) = 0. Further,
he assumed that there was a real-valued function'/(i) tending to 0 as t tends to
0, and such that (A,C) < f(s) if (Á,Â) < å and (B,C) < å. Such a structure
is now called a metric space, although the definition is streamlined somewhat, the
third property being replaced by the triangle inequality. It can be shown that for
each distance function introduced by Frechet there is an equivalent metric in the
modern sense.
In 1906 Frechet also gave two definitions of the term compact (for metric spaces)
in the modern sense. In one paper he defined a space to be compact if every infinite
subset of it had at least one limit point. In the other he defined compactness to
mean that every decreasing sequence of nonempty closed sets had a nonempty
intersection. Thus he used both the Bolzano-Weierstrass property and the Heine-
Borel property (which are equivalent for metric spaces).
General topology. The notion of a topological space in the modern sense arose in
1914 in the work of the Youngs and in the work of Felix Hausdorff (1868-1942),
who was at the time a professor at Bonn. HausdorfT's influential book Grundzuge
der Mengenlehre (Elements of Set Theory) was translated into many languages.
(^48) The early papers of W. H. Young and his wife G.C. Young were published under his name
alone, as mentioned in Chapter 4.