Topology coloring

The purpose of this study is to show how topological surfaces are painted in such a way that the colors are borderless but spaced with the lowest color number. That a surface can be painted with at least as many colors as the condition of defining a type of mapping with the condition that it has no fixed point. This mapping is called color mapping and is examined and analyzed in different conditions of space such as compression or overlap, normality or metric, and continuity, etc. Get the desired result. Then, by proving the theorems and the multiple ones, the color number assigned to each mapping with its specific conditions is obtained. It is proved that, except for one exception mentioned in the text, this number increases to at most n + 3, depending on the particular conditions of each space. Where n can also be subject to finite conditions.