Comparison of proper shape and proper shape over finite coverings

Abstract

The theory of proper shape over finite coverings, defined in [7], uses only
finite coverings to compare noncompact spaces. In this paper we investigate
the relations between this theory and the proper shape defined by Ball and
Sherr in [2]. We show that if two spaces have same proper shape they belong
to the same class in theory of proper shape over finite coverings, but the
opposite doesn’t hold in general.