- TOPOLOGY 391
Compactness. Another basic concept of point-set topology is that of compactness.
This concept is needed to make the distinction between being bounded and having
a minimum or maximum. The concepts of compactness, connectedness, and con-
tinuity are used together nowadays to prove such theorems as Rolle's theorem in
calculus.
At least three lines of thought led to the notion of compactness. The first
was the search for maxima and minima of functions, that is, points at which the
function assumed the largest or smallest possible value. It was clear that a sequence
of points xn could always be found such that f(xn) tended to a maximum value;
that was what a maximum value meant. But did the sequence xn itself, or some
subsequence of it, also converge to a point x? If so, it was clear from the definition
of continuity that ÷ must be a maximum or minimum. This property was studied
by the Czech mathematician Bernard Bolzano (1781-1848), who was looking for
a proof of the continuity property discussed above. He showed as early as 1817
(see Manheim, 1964, P- 67) that the continuity property could be made to follow
from the property that a set of numbers that is bounded above has a least upper
bound. He phrased this statement differently, of course, saying that if there is a
property possessed by a function at some points, but not all, and that property
holds for all points less than some a, there is a smallest number U such that the
property holds for all numbers less than U. Bolzano proved this fact by repeated
bisection of an interval such that the property holds at the lower endpoint but
not the upper. Some 50 years later, after defining real numbers as sequences of
rational numbers (with a suitable notion of equivalent sequences), Weierstrass used
arguments of this type to deduce that a bounded sequence of real numbers has a
convergent subsequence. This theorem, in several closely equivalent forms, is now
known as the Bolzano- Weierstrass theorem.
The second line of thought leading to compactness was the now-familiar dis-
tinction between pointwise continuity and uniform continuity. This distinction was
brought to the fore in the mid-1850s, and Dirichlet proved that on an interval [a, b]
(including the endpoints) a continuous function was uniformly continuous. He was
really the first person to use the idea of replacing a covering by open sets with a
finite subcovering. The same theorem was proved by Eduard Heine (1821-1881) in
1872; as a result, Heine found his name attached to one form of the basic theorem.
The third line was certain work in complex analysis by Emile Borel (1871-1956)
in the 1890s. Borel was studying analytic continuation, whereby a complex-valued
function is expanded as a power series about some point:
If the series has a finite radius of convergence, it represents f(z) only inside a disk.
However, it enables all the derivatives of f(z) to be computed at all points of the
disk, so that if one forms the analogous series at some point z\ in the disk different
from zo, it is possible that the new series will converge at some points outside the
original disk. In this way, one can continue a function uniquely along a path 7
from one point á to another point b, provided there is such a series with a positive
radius of convergence at each point of 7. What is needed is some way of proving
that only a finite number of such disks will be required to cover the whole curve 7.
The resulting covering theorem was further refined by a number of mathematicians,
n=l