Curves in R3\S1
The pictures below show some loops in the space X obtained from
R3 by removing a thickened circle. 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
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 can be regarded as a homotopy in
R3 between two loops that lie in X. However, the
coloured circle moves through the brown region that does not count
as part of X, so the picture cannot be regarded as a homotopy in X.