A modified concept of a shape
of a topological space is introduced which allows some basic geometric constructions:
(1) One has a convenient homotopy concept which originates from a cylinder functor.
(2) All inclusions of compact metric spaces are cofibrations. (3) Shape mappings
which agree on the intersection of their counterimages can be pasted together
(existence of push-outs). (4) There exists a singular complex S which has the
same properties for shape mappings as the ordinary singular complex S for
continuous maps. (5) Consequently one has a singular (shape) homology which for
compact metric spaces turns out to be isomorphic to the (shape-theoretically
defined) homotopical homology (in the sense of G. W. Whitehead) and to the
Steenrod-Sitnikov homology.
