Lecture 4 33

In general, given a compact metric space X, for any ε 0, let

N(ε) be the minimum number of ε-balls required to cover X; that

is, the minimum number of points x1, . . . , xN(ε) in X such that every

point in X lies within ε of some xi. This may be thought of as

measuring the average ‘volume’ of an ε-ball, in some sense; the upper

box dimension of X is defined to be

¯

d

box

(X) = lim sup

ε→0

log N(ε)

log 1/ε

.

We take the upper limit because the limit itself may not exist. The

lower box dimension is defined similarly, taking the lower limit in-

stead. These notions of dimension do not behave quite so nicely as

we would like in all situations; for example, the set of rational num-

bers, which is countable, has upper and lower box dimension equal to

one.

There is a more effective notion of Hausdorff dimension, which

eliminates the need to distinguish between upper and lower limits,

and which is equal to zero for any countable set; because its definition

requires an understanding of measure theory, we will not discuss it

here. For ‘good’ sets all three definitions coincide, and are central

to the study of fractal geometry; however, they are not topological

invariants, so our claim in the last section must be understood to

apply only to a strictly topological notion of dimension.

d. Geodesics. When we are interested in a metric space as a geo-

metric object, rather than as something in analysis or topology, it is

of particular interest to examine those triples (x, y, z) for which the

triangle inequality becomes degenerate, that is, for which d(x, z) =

d(x, y) + d(y, z).

For example, if our space X is just the Euclidean plane R2 with

distance function given by Pythagoras’ formula,

d((x1, x2), (y1, y2)) = (y1 − x1)2 + (y2 − x2)2,

then the triangle inequality is a consequence of the Cauchy-Schwarz

inequality, and we have equality in the one iff we have equality in the

other; this occurs iff y lies in the line segment [x, z], so that the three

points x, y, z are in fact collinear.