over the real or complex numbers, the blowup has a topological description as the connected sum formula_14. assume that "p" is the origin in a ⊆ p, and write "l" for the line at infinity. a \ {0} has an inversion map "t" which sends ("x", "y") to ("x"/(|"x"| + |"y"|), "y"/(|"x"| + |"y"|)). "t" is the circle inversion with respect to the unit sphere "s": it fixes "s", preserves each line through the origin, and exchanges the inside of the sphere with the outside. "t" extends to a continuous map p \ {0} → a by sending the line at infinity to the origin. this extension, which we also denote "t", can be used to construct the blowup. let "c" denote the complement of the unit ball. the blowup "x" is the manifold obtained by attaching two copies of "c" along "s". "x" comes with a map π to p which is the identity on the first copy of "c" and "t" on the second copy of "c". this map is an isomorphism away from "p", and the fiber over "p" is the line at infinity in the second copy of "c". each point in this line corresponds to a unique line through the origin, so the fiber over π corresponds to the possible normal directions through the origin.