A Note on Topology Preservation in Classification, and the Construction of a Universal Neuron Grid

It will be shown that according to theorems of K. Menger, every neuron grid if identified with a curve is able to preserve the adopted qualitative structure of a data space. Furthermore, if this identification is made, the neuron grid structure can always be mapped to a subset of a universal neuron grid which is constructable in three space dimensions. Conclusions will be drawn for established neuron grid types as well as neural fields. Topology is one of the basic branches of mathematics. It is sometimes also referred to as qualitative geometry, in a way that it deals with the qualitative properties and structure of geometrical objects.