  A new methodological approach for the study of topology for shapes made of
arrangements of lines, planes or solids is presented. Topologies for shapes are
traditionally built on the classical theory of point-sets. In this paper,
topologies are built with shapes, which are formalized without points, and with
structures defined from their parts. An interpretative, aesthetic dimension is
introduced according to which the topological structure of a shape is not
inherited from an ambient space but is induced based on how its appearance is
interpreted into sets of parts. The proposed approach provides a more natural,
spatial framework for studies on the mathematical structure of design objects
in art and design. More generally, it shows how mathematical constructs (here,
topology) can be built directly in terms of objects of art and design, as
opposed to the more common opposite approach where objects of art and design
are subjugated to canonical mathematical constructs.
