  A topological space is introduced in this paper. Just liking the plane, it's
continuous, however its $n+1$ regions couldn't be mutually adjacent. Some
important phenomenon about its cross-section are discussed. The geometric
generating element of the coloring region-map is also an important concept.
Every $n$-coloring region map is in the cross-section set of an $n$-color
geometric generating element. The proof of four color theorem and Hadwiger's
conjecture is obtained by researching them and their cross-sections. And we can
see in the context, that those conjectures are not of graph theory, but such
topological space.
