Chapter 9 - Absolute Neighborhood Retracts and Shape Theory

Abstract

Absolute neighborhood retracts (ANR's) and spaces having the homotopy type of ANR's, like polyhedra and CW-complexes, form the natural environment for homotopy theory. Homotopy-like properties of more general spaces (shape properties) are studied in shape theory. This is done by approximating arbitrary spaces by ANR's. More precisely, one replaces spaces by suitable systems of ANR's and one develops a homotopy theory of systems. This approach Hnks the theory of retracts to the theory of shape. It is, therefore, natural to consider the history of both of these areas of topology in one article. A further justification for this is the circumstance that both theories owe their fundamental ideas to one mathematician, Karol Borsuk. We found it convenient to organize the article in two sections, devoted to retracts and to shape, respectively.