John Stillwell - Naive Lie Theory.pdf - Index of

8.6 Paths and path-connectedness 173
Exercises
The above proof of uniform continuity is complicated by the possibility that K is
at least two-dimensional. This forces us to use triangles and the triangle inequality.
If we have K =[0,1] then a more straightforward proof exists.
8.5.1 Suppose that N δ(P1 )(P 1 ) ∪ N δ(P2 )(P 2 ) ∪···∪N δ(Pk )(P k ) is a finite union of
open intervals that contains [0,1].
Use the finitely many endpoints of these intervals to define a number δ > 0
such that any two points P,Q ∈ [0,1] with |P − Q| < δ lie in the same
interval N δ(Pi )(P i ).
8.5.2 Deduce from Exercise 8.5.1 that any continuous function on [0,1] is uniformly
continuous.
8.6 Paths and path-connectedness
The idea of a “curve” or “path” has evolved considerably over the course
of mathematical history. The old term locus (meaning place in Latin),
shows that a curve was once considered to be the (set of) places occupied
by points satisfying a certain geometric condition. For example, a circle is
the locus of points at a constant distance from a particular point, the center
of the circle. Later, under the influence of dynamics, a curve came to be
viewed as the orbit of a point moving according to some law of motion,
such as Newton’s law of gravitation. The position p(t) of the moving point
at any time t is some continuous function of t.
In topology today, we take the function itself to be the curve. That is,
a curve or path in a space S is a continuous function p : [0,1] → S .The
interval [0,1] plays the role of the time interval over which the point is in
motion—any interval would do as well, and it is sometimes convenient to
allow arbitrary closed intervals, as we will do below. More importantly,
thepathisthefunction p and not just its image. A case in which the
image fails quite spectacularly to reflect the function is the space-filling
curve discovered by Peano in 1890. The image of Peano’s curve is a square
region of the plane, so the image cannot tell us even the endpoints A = f (0)
and B = f (1) of the curve, let alone how the curve makes its way from A
to B.
In Lie theory, paths give a way to distinguish groups that are “all of a
piece,” such as the circle group SO(2), from groups that consist of “separate
pieces,” such as O(2). In Chapter 3 we showed connectedness by