The Naive Conception of the Topology of the Surface of a Body

Abstract

The aim of this paper is to sketch aspects of the analysis of some ‘naive’ topological concepts which enter into everyday descriptions of features of the surfaces of physical bodies. Examples of such descriptions and of the reasoning involving them are to say that: (a) Feature A (say a reddish region on the surface of a cube) touches but does not completely cover feature B (say an edge of the cube), from which it can be inferred that feature B touches the edge or boundary of feature A. (b) Several features meet in a common point, from which it can be inferred that any two of them touch, (c) The surface of a transparent object has places or points on it where there are no features (at any rate, no visually discernible ones). It is evident that not only are such concepts as ‘point’ and ‘boundary’ used in everyday descriptions, but they also are assumed to conform to certain general principles which underlie inferences. These general principles can be regarded as elements of an implicit ‘naive topological theory’ of the surface of a body, which I hope to clarify in the present analysis.

The present summarizes in very condensed form results of research carried out over several years. This research has been supported in part by grants from the National Science Foundation (1965–66) and a Guggenheim Fellowship (1967–68). I have benefited from comments of many colleagues, and in particular from extensive discussions with Professor David Shwayder. Lack of space precludes detailed discussion of the relation between the present work and related work of others, and in particular with Shwayder’s (as yet unpublished) work on the analysis of the concept of a body.