Application and development of overlapping space (covering space)

An important concept in algebraic topology is also called covering space. Let P →X be a continuous mapping. If every point X in X has an open neighborhood U, so that p- 1(U) is the union of a set of disjoint open sets {Uα} in X, and P is limited to homeomorphism from Uα to u on each Uα, then P is called an overlapping mapping.

For example, the mapping p:E 1→s 1 from a specified line to the circumference is an overlapping mapping. Let a positive number be taken as the open neighborhood of z0, then p_ 1(U) is the union of a set of disjoint open intervals {(n+t0-ε, n+t0+ε)}, and p:(n+t0-ε, n+t0+ε)→U is homeomorphism. For another example, when each pair of radial points of N-dimensional spherical surface Sn are bonded, the quotient space is a real projective space Pn, and the bonding graph p:Sn→Pn is also an overlapping graph.

This is the overlapping space.