Topologically Fragmental Space and the Proof of Hadwiger's Conjecture

Abstract

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.