Functions. Limits and Continuity. Arcs and Curves 163Proof As before, suppose that 11: z = z 1 (t) and 12 : z = z 2 (t), a:::; t:::; b.
It suffices to let
(3.15-4)
The function ry(t, >.) so defined is clearly continuous in (t, >.), and for each
>. the arc ry(t, >.) = 'f/>., a :::; t :::; b, has a graph contained in E (by the
convexity of E). Also,ry(t, 0) = z1(t) and ry(t, 1) = z 2 (t)
so the arcs /1 and /2 are homotopic in E. The homotopy defined by (3.15-4)
is called a linear homotopy.
Obviously, if the arcs /l and 12 are smooth, each arc T/>. is also smooth.
If /1 and 12 have the same endpoints z 0 and z*, then the homotopy has
those points as fixed endpoints, since
ry(a, >.) = (1 - >.)zo + >.zo = zo
and
ry(b, >.) = (1 - >.)z + >.z = z*
Also, if the arcs /l and 12 are (closed) curves, so is each 'f/>., smce
z1(a) = z1(b) and z2(a) = z 2 (b) implies that
ry(a, >.) = (1 - >.)z1(a) + Az2(a) = (1 - >.)z1(b) + >.z2(b)
= ry(b, >.)we may also note that in the case of a linear homotopy, the trajectories are
line segments, as follows from (3.15-4) with fixed t.
Theorem 3.15 Let 1': z = z(t') = z(h(t)), a:::; t:::; b, be a reparametriza-
tion of the arc 1: z = z(t), a :::; t :::; b, where h(t) is a continuous,
real-valued, nondecreasing function with h( a) = a and h( b) = b. Then
/^1 is homotopic to / ·
Proof A homotopy between 'Y and -y' can be defined as follows:
ry(t, >.) = z((l - >.)t + >.h(t)),
clearly, for >. = 0 we get ry(t, 0) = z(t), and for >. = 1 we get ry(t, 1) =
z(h(t)). The two arcs have the same endpoints, since z(h(a)) = z(a) and
z(h(b)) = z(b).
This theorem shows that the homotopy class of an arc is determined by
the order of succession of the points on its track rather than by a particular
defining equation.
Definition 3.29 An open set A is said to be a simply connected if it is
connected and if every curve with graph in A is homotopic to a point.