The surface of the earth (E), a region (Rm), and its components (e) should be defined in the set theory as follows,
E⊃Rm, Rm∋e, β=φe, φe=μφn
on condition that e=sociotop, β=social function,
φ=function of e, μ=function of n, and
n=natural materials of e, (physiotop and ecotop).
These expressions come out only through the behavior of a social group in Rm. There needs Rm→Rmn in order to consider Rm a complex of various behavior spaces composed of e-group.
fa: Rmn→Va|V=2-dimentional cross section of a complex of behavior spaces
f-1: {Va, Vb, ……, Vo}→Rmn
while it is only the valuable e-group in the actual field of activities that appear in the presence, some other e1, e2, ……disappear in the back. In process of action, en in focus takes turns and same one changes its scale and outline. From the facts that e1, e2……happen to turn inside out on one's return, we must consider still more that a behavior space is often transformed into a projective plane
ha: Va→V'a|h=projective mapping
∴ hafa: Rmn→V'a (1)
h-1f-1: {V'a, V'b, ……, V'o}→Rmn
When a means of transportation is on solid crossing, Rm becomes homeomorphic with a torus.Plane surface ACEFDB would be homeomorphic with a disk, if there were a means of transportation to connect A with C and E, B with D and E. But if a new multistory highway is constructed, the curved surface would be homeomorphic with a torus. More, suppose that a connection of A, D, F and of B, C, E is strengthened, a Möbius band's projective plane comes into existence. μ index, μ=e-v+p in graph theory, as well as value of Auler index, X(F)=v-e+f in topology, must change under high dimentional conditions of each curved surface.
In the basic model according to the marketing principle postulated by W. Christaller, a service area of each central place corresponds to 2-dimentional plane formed by a dual graph of a planer graph which vertexes coincide with central places of the same order. But a network presented on the basic model ought to be considered to over 2-dimention, because it consists of a set of planer graph and daul graph. The new model of a service area is hier explained to take up the mechanism of s-dimentional manifold.
If Rmn were to be supposed to keep a certain balance, it is impossible to pick up a cross section (V) as a net of hexagonal-shaped pattern (G), which is cut by Rmn. At least there needs a projective transformation,
ε: V→G (2)
The relation between (1) and (2) might be explicated after the fact that this transformation proves to be correct in projective spaces of some regions. A part of Rmn's deeper structure is exposed by way of example of the uneven boundaries of regions based on the catastrophe theory of topology.
