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 Ł 1
	=	(x,y)/r	if r ł 1
h(t,x,y)	=	(1-t) (x,y) + t g(x,y)

The following picture illustrates the map g.