The basic idea behind Borsuk’s shape theory is explained in the entry on shape theory, so will not be repeated here, except to say that it considers compact metric spaces embedded in the Hilbert cube, then uses the open neighbourhoods of the space as a ‘net’ of approximations of the space. The space is, of course, the intersection of all these open neighbourhoods.
Any compact metric space can be embedded in the Hilbert cube, so it is sufficient to consider just compact subspaces of that space.
Let be the pseudo-interior of the Hilbert cube, .
We will define (a category equivalent to) the Borsuk Shape category, , to have compact subsets of as objects and some morphisms that need a bit of explaining.
If and are compact subsets of , then a fundamental sequence, , is defined to be a sequence of maps such that for every neighbourhood of in , there exists a neighbourhood of in and an integer such that if , the restrictions and are homotopic within .
Note that the do not have to be contained in , they only have to be ‘near’ .
Two fundamental sequences, , are said to be homotopic, provided that for every neighbourhood of in , there is a neighbourhood of in and an integer such that if , then and are homotopic within .
The morphisms of and taken to be the homotopy classes of fundamental sequences between the corresponding spaces.
Two compacta contained in are said to have the same shape if they are isomorphic in . As an example, the Warsaw circle has the same shape as the circle.
If and are compacta in , then and have the same shape if and only if their complements and are homeomorphic.
Chapman extended the association ‘goes to’ to a functor from the Borsuk shape category to the weak proper homotopy category of complements in of compacta. This was the basis for Edwards-Hastings formulation of strong shape theory, on replacing the weak form of proper homotopy by a strong form.