The plane

The plane itself is contractible, or in other words, homotopy equivalent to a single point:
h(t,x,y) = (tx,ty)

The plane with one point removed is homotopy equivalent to a circle.
h(t,x,y)=(1-t+t(x2 + y2)-1/2)(x,y)

If we remove two points, the remaining space is homotopy equivalent to the union of two circles that meet at a single point (or in other words, a figure eight).
r = (x2 + y2)/|x|
s = x/|x|
t = ((x-s)2 + y2)1/2
g(x,y) = (x-s,y)/t + (s,0) if r 1
= (x,y)/r if r 1
h(t,x,y) = (1-t) (x,y) + t g(x,y)

The following picture illustrates the map g.