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(x^{2} + y^{2})^{-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).