Proper $CW$-complexes: A category for the study of proper homotopy

The notion of closure finite complexes with weak topology introduced by J.H.C. Whitehead determines an adequate category for the study of homotopy theory. Nevertheless a noncompact space which can be described as a $CW$-complex always needs an infinite number of cells. In the present paper we develop a new notion that we call proper $CW$-complex which enables us to describe some noncompact spaces with finitely many cells.\newline In this new category we give an algorithm which permits to compute the singular homology groups and the proper homology proups associated with a finite regular proper $CW$-complex. On the other hand, a characterization of the proper homotopy equivalences depending on the Hurewicz and the relative Steenrod groups is obtained. The paper finishes giving some results about proper cellular approximations of proper maps between proper $CW$-complexes.