Odds and ends
The space R\Z is homeomorphic to a sisjoint union of
infinitely many copies of the interval (0,1). Equivalently, it is
homeomorphic to Zx(0,1), and thus homotopy equivalent to
Z.
Here is a picture of the effect of the map
e(t) = exp(2p it). On the
left, we see it as a map from C to C\{0}; on the
right, we see it as a map from R to S1.
Here is a curve moving in the complement of a trefoil knot:
Here are the simplices of dimension 0, 1, 2 and 3. You can rotate
them with your mouse.