Curves in a plane region Let X be the subspace of the plane
R2 shown in black in the pictures below. The
pictures illustrate some homotopies between loops in X. Recall that
a loop is a continuous function u:[0,1] -> X such that u(0)=u(1).
The points on the loop are colour-coded: the reddest point is
u(0)=u(1), the blue-green point is u(0.5), the yellow point is
u(0.2), and so on.
The first picture shows a homotopy between two curves.
The second picture shows another homotopy between two loops, say u
and v. Here the images of u and v are the same set, but the
colourings are different, which means that u and v are different
functions and so count as different loops.
The third picture is a reminder that he map u:[0,1] -> X need not be
injective, or in other words that a loop is allowed to cross over
The final picture shows two loops (say u and v) and a homotopy
between them, where the intermediate stages are non-closed curves.
Thus u and v are homotopic as curves, but they are not homotopic as