Abstract. We give in this paper a description of proper shape theories of arbitrary topological spaces. Our method is to use multi-valued functions with smaller and smaller images of points. An analogous intrinsic approach to shape theory of compact metric spaces was earlier considered by J. Sanjurjo. The author has extended it to arbitrary topological spaces and the present paper shows how this extension can be adapted to the proper case. The main result is a construction of the proper {shape} category ${\cal S}h_p$ whose objects are topological spaces and whose morphisms are proper homotopy classes of proper multi-nets. The category ${\cal S}h_p$ relates to the proper homotopy category ${\cal H}_p$ similarly as the shape category ${\cal S}h$ links to the homotopy category $\cal H$. On compact spaces the proper shape category agrees with the shape category. For locally compact metrizable spaces, we show the existence of a natural functor from our proper shape category into Ball's proper shape category ${\cal S}^1_p$ which is similarly related to the original Ball and Sher proper shape category.

AMS Subject Classification (1991): 54B25, 54F45, 54C56

Keyword(s): proper multi-valued function, {\sigma }, -{close}, {\sigma }, -{small}, {\gamma }, proper-{homotopy}, proper multi-net, proper shape theory, trivial proper shape, proper shape equivalence

Received December 22, 1993. (Registered under 5614/2009.)