Topological Foundations of Cognitive Science

1. The Concept of Transformation A first introduction to the basic concepts of topology takes as its starting point the notion of transformation. We note, familiarly, that we can transform a spatial body such as a sheet of rubber in various ways which do not involve cutting or tearing. We can invert it, stretch or compress it, move it, bend it, twist it, or otherwise knead it out of shape. Certain properties of the body will in general be invariant under such transformations which is to say under transformations which are neutral as to shape, size, motion and orientation. The transformations in question can be defined also as being those which do not affect the possibility of our connecting two points on the surface or in the interior of the body by means of a continuous line. Let us provisionally use the term 'topological spatial properties' to refer to those spatial properties of bodies which are invariant under such transformations (broadly: transformations which do not affect the integrity of the body or other sort of spatial structure with which we begin). Topological spatial properties will then in general fail to be invariant under more radical transformations, not only those which involve cutting or tearing, but also those which involve the gluing together of parts, or the drilling of holes through a body, or the decomposition of a body into separate constituent parts.