The History of Mathematics: A Brief Course

QUESTIONS AND PROBLEMS^393

The first part of the book is an exposition of abstract set theory as it existed at
the time, including cardinal and ordinal numbers, and the early stages of what is
now called descriptive set theory, that is, the classification of sets according to their
complexity, starting with a ground class consisting of closed sets and open sets, and
then proceeding up a hierarchy by passing to countable unions and intersections.
He invented the term ring for a class of sets that was closed under finite unions and
intersections and field for a class that was closed under set differences and finite
unions, but warned in a footnote that "the expressions ring and field are taken
from the theory of algebraic numbers based on an approximate analogy that it will
not do to push too far."^49
Hausdorff introduced metric spaces, being the first to use that name for them,
via the axioms now used, then gave a set of "neighborhood axioms" for a more
general type of space:

1. To each point ÷ there corresponds at least one neighborhood Ux; every
   neighborhood Ux contains the point x.


2. If Ux and Vx are two neighborhoods of the point x, there is another neigh-
   borhood Wx of x contained in both of them.


3. If the point y lies in Ux, there is a neighborhood Uy contained in Ux.


4. For any distinct points ÷ and y, there are two neighborhoods Ux and Uy
   whose intersection is empty.
   These were Hausdorff's axioms for topology, and they were well designed for
   discussing the local behavior of functions on a highly abstract level. A quarter-
   century later, the group of French authors known collectively as Nicolas Bourbaki
   introduced a global point of view, defining a topological space axiomatically as we
   know it today, in terms of open sets. The open sets of a space can be any collection
   that has the empty set and the whole set as members and is closed under arbitrary
   unions and finite intersections. In those terms, one of Hausdorff's neighborhoods
   Ux is any open set Ï with ÷ G O. Conversely, given a set on which the first three
   of Hausdorff's axioms hold, it is easy to show that the sets that are neighborhoods
   of all of their points form a topology in the sense of Bourbaki. Bourbaki omitted
   the last property specified by Hausdorff. Spaces having this extra property are now
   called (appropriately enough) Hausdorff spaces.

Questions and problems

12.1. Judging from Descartes' remarks on mechanically drawn curves, should he

have admitted the conchoid of Nicomedes among the legitimate curves of geometry?

12.2. Prove Menelaus' theorem and its converse. What happens if the points Å and

F are such that AD : AE :: BD : 偼 (Euclid gave the answer to this question.)

12.3. Use Menelaus' theorem to prove that two medians of a triangle intersect in

a point that divides each in the ratio of 1:2.

(^48) The word ring in the abstract algebraic sense was also introduced in 1914, in a paper of A.
Fraenkel (see Section 2 of Chapter 15). A very influential work in measure theory, written in 1950
by Paul Halmos (b. 1916), caused Hausdorff's ring to fall into disuse and appropriated the term
ring to mean what Hausdorff called a field. Halmos reserved the term algebra for a ring, one of
whose elements was the entire space. Probabilists, however, use the term field for what Halmos
called an algebra.